This doesn't contain any information about the game walkthrough, so I recommend initially doing it yourself if you haven't already. For the most part, the game doesn't provide any difficulty in normal playthrough, except for a few moments honed to your reaction and speed of action, so you won't have any problems.
Also
Game has 1 achievement that require 3 of you friends with a purchased copy of the game on Steam, or having 4 controllers to play local multiplayer and 1 achievement require having at least 2 controllers .
But!

1. I'm not sure if the online multiplayer works properly in this game.
   When trying to play 3 of us with our friends, we simply couldn't start the game. All players except the host stayed in the lobby while the host started the game alone. Perhaps online multiplayer works only when there are 4 players, perhaps it doesn't work at all. If anyone has managed to play online - report about it in the comments.
   
2. Local multiplayer works correctly, but you will need 4 controllers to get one of the achievements.
   
3. If you (like me) don't have 4 controllers, you can still get achievements using third party software to emulate them. You can read detailed instructions on how to set this up in the “controller emulator” section.

You need to collect all the cheese to access the game mode you need for the achievement.
Once you have picked up a cheese, it instantly counts towards the found cheese list, you don't need to go through the chapter to the end or play until the next checkpoint if you have no uncollected cheeses in the future.

Will meet you on your way through.
Located to the right of where you first went up with the ball. You can fly over the fence with an ice jump or break the wall using ice and jump to it.
Left on the pipe. Wait a bit until the platform turns higher towards the cheese and then jump up using the ice and then pull the cheese towards you or climb the wall using the ledges and then jump up using the ice to climb directly onto the pipe.
Top right. Climb up the inclined platform and then go back to pick it up.
It meets in your way at the end of chapter. Just pick it up before you jump off.

At the start of the chapter. Jump over the chasm and go into pipe.
Break the wall in this area using ice or kill bomb-enemy there.

Press the phone number 4, 7 or 9.

On the right side of the board. It looks transparent on the screenshot, so you may not notice it in the game. Pull him to you by jumping on the barrier, or fly to him with the help of ice.

Left on the platform. Pull him towards you or swap places with him, then die to pick him up on the other side.

Will meet you on your way through.
Lying among the balls, it's not visible in screenshot, but you should see it before you go to this place.
From the left side of the playing field. The camera will pass near it when you shoot the ball into the goal.

Jumping onto the conveyor belt wait a little while for him to appear.
Top left. Use the ice to jump higher, then pull the cheese towards you. Be careful not to drop the cheese into the lava, you can sacrifice your life to get it, otherwise you will have to replay the chapter up to this point again.
To the right of the raft. The screenshot is late, but you won't be able to see it there anyway. So when you get to the raft, you need to teleport immediately to the place of the cheese and then die to revive and pick it up.
Upstairs. You need to place an ice block, stand on it, then jump up and pull the block towards you to push it again to fly even higher, and then you need to swap places with the cheese.
Press any piano key to get cheese.

This section is only for those who want to get all the achievements alone, without using physical controllers.
I can't guarantee that this software is safe but it works and is quite popular, if you have other options you can use them. You install it and additional components at your own risk.
So, we are going to use x360ce. Also, instructions can be read on website, however, I'll post about it here as well, in the proper sequence.To do this we need to go to the official site https://www.x360ce.com/ and install the ‘for all games’ version.
Unzip the .exe file and run it.

Next, you need to check if the associated driver is available.
This screenshot outlines everything we'll need:
1. Select a controller.
2. Enable/disable controller.
3. Menu for selecting the device from which we will emulate the controller.
4. Button for adding a new player
5. Buttons for selecting a team in Tiny Soccer mode.

What to do:
1. Select controller 1 and click Enables 0 Mapped Devices.
2. Now select controller 2.
3. Click on Add button, in the new menu select your keyboard as input device by clicking on ‘Add Selected Device’.
4. Now you can configure to assign keys to your controller, click on the desired one, select record and bind any convenient key.
5. Back in the game press the newly bound key to add a new player. If you are playing Tiny Soccer you will also need to select team.
6. If you need to add more than one player, repeat steps 2-5 for controller 3 and 4.

Other:

• You can only use one controller bound to your keyboard at a time.
  
• Based on previous fact, you can bind same key for the ‘A’ key, also when you switch devices emulator saves previous key, so you may not select it at all, and skip step 4.
  
• It is quite realistic to play Tiny Soccer mode if you have a friend with whom you share keyboard, but rotating camera using keyboard will not be a pleasant experience, otherwise you will still need 2 physical controllers.

Welcome, Tiny Brains!
Started the experiment!


You'll automatically get it when you start a story game mode.

Intelligence Level: Tolstoi
Defeat the final boss in "Escape The Mad Lab" mode.


You will automatically get it by defeating the boss and Pinky in singleplayer story game mode.
Maybe this achievement is given with ‘Intelligence Level: Kamarad’ if it was your first playthrough, can't check that. If not, you don't need to go through all the chapters in single player, just start chapter 4 and complete it.

Intelligence Level: Kamarad
Defeat the final boss in "Escape The Mad Lab" with 3 other players.


You will automatically get it by defeating the boss and Pinky in multiplayer story game mode with 3 other players.
Also like with "Intelligence Level: Tolstoi" you don't need to go through all the chapters in multiplayer, just start chapter 4 and complete it.
If you want to get this achievement yourself, you need to use emulator(read the instructions in the corresponding section):

1. Start chapter 4 in offline mode, go through chapter and defeat the boss alone, but don't rush to kill Pinky.
   
2. At this point you need to alternately add 3 ‘players’ using the emulator.
   
3. Turn on 2 controllers, then select second controller, press key that is responsible for the ‘A’ for the current controller and second player will appear.
   
4. Repeat this action twice more, adding new controllers until you have all 4 characters in the game.
   
5. After that just jump on Pinky.

She will disappear and chapter will be finished. In my case, achievement was not awarded immediately after jumping, only after cutscene started.

Intelligence Level: Zaytsev
Complete any 'Protect' combat sequences without the VIP getting hit by enemies.


Kill all enemies so that defended object does not take damage.
You will most likely get this achievement during your normal playthrough. If not, I recommend doing it in chapter 2 when you first encounter it, or in chapter 4 when you need to protect the professor, as I find them the easiest.

Intelligence Level: Rasputin
Defeat the final boss in Tiny Trolls mode.


Doesn't support online multiplayer.
You don't need to complete all chapters. Just start chapter 4 and complete it.
Tiny Trolls mode changes the appearance of the characters and includes ‘friendly fire’. If you play in single player, there will be no changes except for the appearance.

Intelligence Level: Completionist
Complete all Chapters in Jules Mode.


You need to complete all chapters. It can be done over several game sessions, the main thing is that when you select a chapter, you should have a previous record of passing it for time.
Jules Mode can be played exclusively in singleplayer. In this mode, abilities of all characters have been merged, so you no longer have to switch between them. In addition, you only have one life, if you die for any reason, you will have to start the chapter from the beginning. This does not affect the loss of a ball or the death of a defended object, in these cases you will return to last checkpoint.

Intelligence Level: Gretzky
Finish 1 match of Tiny Soccer mode.


This mode only works in local multiplayer. You will need at least 2 controllers to get the achievement.
If you are using an emulator, you will need to:

1. Enable two controllers.
   
2. Under control of controller 2, press ‘A’ to add a player.
   
3. Press ⭢ or ⭠, depending on which team you want to join (using the keys you assigned).
   
4. Start the game using the keyboard.
   
5. Just wait until time runs out.

I did all achievements alone, maybe if you do them with two or more players they will be easier.
It should also be clarified that about half of Challenges are only unlocked after you accumulate a certain number of stars. For this you will have to play other challenges that don't have achievements.
In my case, the easiest (3 stars) were levels related to solving puzzles for speed, a little more difficult levels where you need to drag a ball to hole in a certain time (I got 3 stars, but it takes practice, I'm sure that for 10-20 minutes you can cope without any problems). Then there are levels where you need to protect the object, 2 stars is not that hard to get playing alone, for 3 it takes a bit of practice, i think if you play with two people it will be very easy.


Intelligence Level: Pavlov
Score 1200 cm or higher in The Ball Washer.


I spent a lot of tries to get this, one of the hardest challenges probably.
My recommendations:

• Don't use a lot of push after a distance of 300-400m, it's very dangerous.
  
• Keep using the pull-swap tactic.
  
• If there is a chasm ahead or too much glass stop the ball with ice to wait for a safer route.
  
• If there is enough safety ahead, you can push the ball forward, but be prepared to react quickly and swap/attract it in case of danger.

On average, it will take you about 2-3 minutes to reach this distance.

Intelligence Level: Gagarin
Complete The Déjà Vu Puzzle in less than 10 seconds.


Use the solutions from these videos(not mine):
https://youtu.be/WpWDs8D0epQ?si=dC28Br5YGvVae-xZ
https://youtu.be/k1Sr5OcEFNk?si=1Tvuerq41ZHH50zi


Intelligence Level: Vladimir
Survive for 80 seconds in The Ball Carousel.


Recommendations:

• Always keep ball as far to the left as possible.
  
• The push is too risky, only use it when ball is too low and you need to lift it up, but be careful.
  
• Try to keep the ball on ice at all times to prevent it from rolling away.
  
• As with ‘The Ball Washer’, pull or use pull-swap tactics to immediately ice the ball at the new point.
  
• There are situations in this challenge where teleportation is extremely useful, keeping ball safe and not requiring additional ice control (as in the screenshot).

Intelligence Level: Borzov
Score 750 cm or higher in Saucer Ball.


This challenge is similar to The Ball Washer, but easier. Same recommendations, but apart from that I advise you to use the freeze-swap tactic to get forward, it's mostly safe in long distance, but be prepared to attract ball in case of danger.


Intelligence Level: Tesla
Finish The Glass Puzzle.


I won't explain how to get through this, as you have no time limit and you don't need to get a star to achieve it. Just treat it as an extra level outside of the story. :)