Are Ainz and his servants still leveling up the same way they were when they were killing monster to get exp in Yggdrasil?
Hmmm... To my understanding, Ainz and the high-tier minions (read those that are level 100) don't level up any further, as they are maximum level already. It's never mentioned that they 'get stronger' while others like the lizardmen and that weird hamster indeed do (the narrative comments several times that they are training to get stronger). Now, it is possible the New World has a higher level cap but insofar it hasn't been mentioned at all in the nine LN, the WNs nor the anime and manga. So Ainz ain't levelling nor are the floor guardians, but some of his minions do.
Lvl 100 was maxed level in the game, but the New world has different laws and something from the game manifest differently. This is because the new world isn't a game, but instead, an actual world that is trying to make sense of the game mechanics.
In the game, the minimum amount of experience a monster could give is 1 and also Ainz had 100% experience to level if the game allowed him to become level 101.
Ainz should have leveled already if he was gaining experience points, the only caveat is that what if the New world is giving humans or other monsters less experience. For example, if the new world gives humans 0.001 experience then it could still be possible, but there isn't anything that suggests that Ainz can level.
Last note, the author himself said Ainz is maxed level and he won't be able to level.
Reference to the translation, which has a reference to the original.
With regards to the Denizens of Nazerick leveling it's hard to tell. Obviously as the author has said, the level 100 characters have reached their level cap and are subsequently unable to level.
The more interesting question is wether the sub-lvl.100 characters are able to level (i.e. the Pleiades and the area-guardians). It is difficult to be sure wether or not these characters are able to level since we don't often see through their perspective and they (with the exception of Narberal) haven't had a lot of combat experience yet.
We know that Players from Yggdrasil can level as evidenced by the strongest of the "13 Heros," who came to the New World between lvl.10-20, started out as the weakest of the heros, and ended as the strongest of them. We can also be fairly confident that the New World Natives can level (albeit, likely with a racial/personal level cap, as evidenced by the fact that the Godkin are technically a different race and are much stronger than normal humans), due to the existence of Adventurers, humans who, through combat, have become stronger than the average human. Also the Nazerick trainees, (i.e. the Lizardmen and Hamusuke, since it's been stated that they're getting stronger)
The real question is wether or not the NPCs can level. There isn't a lot of info on it, but I'm fairly certain that NPCs were not able to level naturally in Yggdrasil, if that carried over into the New World then whatever level they were when they transitioned from Yggdrasil is effectively their level cap. It's difficult to say wether or not this is the case even if they could grow to lvl.100. As I mentioned earlier the Denizens of Nazerick haven't really had any challenging battles so far and they haven't been doing any leveling/grinding, which at their levels and considering the relative weakness of the New World, they would need to do quite a bit of in order to see any real results.
Personally I don't think that any of the original Denizens of Nazerick can level in the New World. However, that does leave one avenue for them to still get stronger, that being Martial Arts and new Spells.
Though leveling up does provide a stat boost, it appears that one of the primary ways it increased players strength back in Yggdrasil, especially at higher levels, was through the gaining and improving of skills/magic. If Ainz and the Denizens of Nazerick are able to learn new Spells and/or Martial Arts it could provide a lot of room for growth, especially at their levels, considering that most of these abilities seem to scale with the user's strength. Though frustratingly there hasn't been any development on that front because none of them are actually attempting to learn anything new (mostly because they're all already strong enough to whup just about anybody's ass in the New World) The closest they get is the Death Knight training with Hamusuke, who is a created undead, so who knows if he even counts?
The closest we get to a confirmation or denial is shortly after Nazerick arrives in the New World. Ainz tries to get one of the maids (a homunculus maid, not one of the Pleiades) to cook a steak, like, three times and the maid fails in the same way each time. If I remember correctly, because of this Ainz assumes that the NPCs can't learn new skills. However this was still while New-Nazerick was in its infancy and the NPCs were just getting used to the New World and its freedoms. It could be that the issue with the maid was one of mindset. I don't belive the LN provides any details on how the steak was cooked, but there's a chance that the maid, being used to Yggdrasil where without a skill in something one could not do that thing, and being entirely new to cooking, didn't know what was going wrong and just did the exact same thing all three times. If she had been instructed on how to cook and had had some more practice, maybe she could have learned.
Though this is all just a bunch of speculation. Even if the Denizens of Nazerick can learn new skills, there likely won't be any effort/development put forward in that department until/if they encounter an enemy who is stronger than they are, (maybe another guild will transfer over from Yggdrasil or something) which definitelywon't be happening in the story for some time yet.
I was reading the books and Ainz is clearly saying he should have leveled up but didn't feel anything different so he concludes he can't level up. But I think he is wrong. He is a better fighter and he is doing things no spell caster would be able to do. Meaning he has obtained some fighter levels.
At smaller "levels" it is easy to see an increase in power but when you have 100 "levels" getting 1 more is not so evident. Also I think that in the new world there is no fix levels. In games you only get skills when you level up but now I think you learn the skill step by step for example at the middle of you leveling from lv2 to lv3 you can do the skill that you gain at lv3 but not as good.
There is an example that goes against this when Neia Baraja uses divine power for her archery it seems like an instant gain unlike Hamsuke constant training and step by step level up.
Of course a reason he would be able to be better at fighting but not level up could be because he is getting used to his powers and learning how the world works. Also gaining power from flavor text with no mechanical gain in game but true now. If that is true then it would look like leveling but only discovering his powers.