When you carry over a monster to NG+, she keeps the following:

• Her name duh
  
• Her maximum stamina and mental
  
• Highest potential level for OBD, LWD, and TEC (up to level 9)
  
• Skills and Prostitution stats (obviously also virginity state)
  
• AFF (affection level)

That being said, there are 2 things that do get reset:

• OBD, LWD, and TEC get reset to default starting level
  
• PRD (pride) and SHM (shame) get reset to max value

Since monster girls carry over stamina, mental, and highest potential OBD, LWD, and TEC from using seeds, it doesn't really matter who you pick for these reasons alone since you can get any monster girl with high stats from just items. Still, there's only so many seeds available in the shop during each playthrough so generally you'll end up picking monsters that start with high potential stats.

Each monster has skills that are either a fetish type for clients, or a passive/active ability etc. Not all skills can be exchanged, we'll get to that shortly.
Skills fall into 4 Categories:

• Body Type: These skills come with the girl by default and cannot be exchanged.
  
• Personality: These skills mostly come with the girl by default, sometimes they get unlocked like with Salamander's Tsundere skill. These skills can be exchanged.
  
• Special Skills: These are skills obtained through training, for example: doing urination or anal training for angel enough times will result in the Wetter and Anal Preference skills respectively.
  
• Active/Passive skills: These don't have any special tags, most of the time they either come with the monster girl by default or are unlocked after reaching a certain AFF or LWD level idk it depends. Usually they can be exchanged.

Exchanging skills is unlocked by buying the building under Construction.

You select two monsters, and you can copy a skill from the monster on the left to the right if the skill is transferable and if you have enough exchange points on the 1st girl.

Exchange points are gained by training the girl, and upgrading the building will let you gain more exchange points during training.

I know I said stats don't matter, but her starting base stats are pretty high anyways. She is also an egg layer so every 8 days you'll get a nice cash injection. She does have the Rare body type skill so that's an automatic increase of 1.5x customer satisfaction too.
These are the skills I have on her:

• Passionate doubles the bonuses from squeezing a client dry from 150% to 300%, and can be transferred from Salamander.
  
• Melody of the Sea: this is the best skill especially once you get upgrades for more clients and extentions packing Bifrost with customers and your girls can't keep up. It resets the cooldown time for all your girls and can be transferred from Kraken.
  
• Pop Idol Song: I actually have no idea what this does but there's nothing else really, you can swap this out if you want for a personality skill.

She's a bit rough to break in because of her starting pride and shame without diuretics, and some would argue she's kinda useless with her long cooldown time as well. However, Angel starts with pretty high potential stats and with these skills, can really milk the money out of wealthy customers, sometimes hitting over 1000% customer satisfaction.
Skills:

• Passionate, same as above get that from Salamander
  
• High Class: there's a bunch of monsters with this, like queen naga and kitsune, also yuki-onna. This just increases customer satisfaction.
  
• Mushroom Spores: Increases energy damage to clients and pleasure for girls in active party, this is pretty nice to have to help drain client energy and max out pleasure bars for more satisfaction bonuses. Can be transferred from the Myconid
  
• Immoral Pollen: Increases Client Satisfaction. Idk if this is just for the girl with the skill or party wide, if the former then you should probably replace pop idol song on queen naga with this.

Dark fairy has pretty ass stats but it doesn't matter, we're after her for her unique skills. Once your affection with her is high enough, you can regain action points from doing penetration training due to her Fairy Secretions skill, which is essential for getting girls ready for escort missions and raising stats. She's pretty fragile so try to keep her on standby after you buy enough space to get her out of the active party, as she's more useful as utility.
Skills:

• Masochist: this is found on quite a few monsters, myconids are pretty cheap and so are mummies. It just helps her out a bit since her training is pretty painful.
  
• Fortune Summoner: Doing penetrative training will increase the likelyhood of wealthy customers appearing that night, which has excellent synergy with Fairy Secretions. This can be transferred from Oni.

You'll want to keep Queen Naga and Angel in the active party pretty much all the time, you'll be making enough money to sustain them with items.

Try to use Mental Refresher EX on your most important/fragile girls since they can still be recovered if they run out of stamina, but become useless if they mentally break. Don't suffer the sad experience of just keeping them around out of pity since you don't want to sell them for 10G after all they did for you.

Doing training with Dark Fairy also results in Fairy Dust, which you should always use to train your other monster girls who don't have the spirit skill. Combine this with aphrodisiacs if you have the money to spare and enough action points to really power train a girl.

If you have the space, you should buy a bunch of Myconids and perform Left Alone training on them for free mushroom items every day.

Anyways, let me know who your favorite monster girls are! I have quite the soft spot for elf and ogre, but really everyone is so darling and has their own charm.