So, I'm sort of in a unique situation here in that while I specialize in PMD, I also have at least a single one shot already written in my "trainer verse" and have a few others in mind as well. So, with that being said, I actually have several mechanics that cross over in my PMD setting of Kilo, and my trainer setting that's more or less a canon interpretation. So, where possible, I'll answer the questions for both!
1. Does the four move limit still exist? Do moves themselves exist?
Yes and no. In Kilo (my PMD setting) Pokemon are capable of learning any number of moves and can, at will, switch them up based on any moves that they had learned previously as long as they remember the aura memory behind it. In other words, maybe they didn't use Iron Tail in a while, but they can give themselves some time (several seconds to minutes) to 'recall' to move, and at that point, recollection becomes much easier.
The "quick recollection" method seems to be an intrinsic property of all auras (with some exceptions for non-mortals, such as Mew) and, essentially, Pokemon are only able to "hotkey" four moves at any given moment, and need extra time to call upon or otherwise 'swap out' any of these four moves.
From a narrative standpoint, my philosophy here (aside from generally preferring to stick to canon) is that limitation breeds creativity, and by limiting characters to only so many moves, it can show a lot about their battle style, personality, occupations, or tendencies by which moves they have on the front lines, and which ones they'd abandoned, neglected, or otherwise benched.
As a concrete example, I have my Trapinch, Gahi, who prefers stealthy or evasive maneuvers, so he has the likes of Sand Attack and Feint Attack on his preferences. Meanwhile, pure offensive fighters like my Axew, Demitri, focus almost entirely on hitting things and hitting them hard, so Dual Chop, etc. And my main character, Charmander Owen, barely has any offensive moves at all at the start of the story, having both Protect and Smokescreen so he can generally stay out of fights or otherwise battle from a distance. Yeah, you can just give them "preferences," but it doesn't quite have the same impact as they simply only being able to use those techniques on the fly.
In the trainer world, it's largely the same, but Pokemon aren't quite as educated about how to easily switch their moves around. Their trainers or mentors or other equivalents instead help them tune to the techniques they need. Examples include the Tutors, TMs, etc.
2. Do types and type advantages still exist?
Yep. Nothing really special to remark here.
Edit: I lied I actually have something special to point out here.
While I typically follow all Type matchups when it comes to Type versus Type, I do have a few stretches of canon now and then to accommodate for environmental factors. The logic is supported partially by canon as well; Ice Types are immune to hail despite Ice being typically neutral against Ice, or a Fire can't be burned even though they aren't 0x immune to Fire.
As an example, my story has a scene right near the start of my Charmander sleeping in fire because it's cozy. In several settings, this would be dangerous, but I decided to roll with this because, particularly in PMD, Fire Types can walk on lava with zero penalty.
On the flip side of things, he also has to be wary of bodies of water, because that's generally very unpleasant. In fact, if his tail gets doused, he runs the risk of going into "water shock," which is the Charmander equivalent of trying to walk after suffering a charlie horse to both legs, so to speak. No, he doesn't die if the tail goes out; the association is reversed--it's an indicator of life, not a provider.
In Kilo specifically, there are actually workarounds for this for quality of life purposes. While I only depicted one particular instance of this, the anti-Super Effective berries have a small world building factor here. my Charmander often uses Passho Berries or Passho Powder to take baths, simply by mixing it into the water. This makes, essentially, Fire-friendly water so he can bathe without any discomfort. I also have later on in the story a Hydreigon and Gardevoir pair; despite the Type issues, a simple Roseli Berry can make close interactions less... risky, so to speak.
3. What is the relationship between trainer and 'mon like in battle? Is it that of a master and servant, or more like a coach and athlete?
Nothing for Kilo here, as humans are pretty much these mythical things that folks don't even know what they look like. But for the trainer world, it's neither coach and athlete or master and servant. My trainer world has non-wild pokemon at extremely high autonomy levels, perhaps even a step further than what the recent Galar entries imply. They can leave their Poke Ball at any time; they typically actively choose to battle alongside or with their human partner.
The entire "trainer" journey is more like a means for children to have a hands-on learning experience with the world and how intrinsic humanity's relationship to Pokemon really is. Very rarely do trainers truly "complete" their Gym missions, so much as use it as a platform to find what sort of career they'd truly want to study for by getting out there in the world.
Okay, I kind of went off on a tangent there, but basically my point is, Pokemon voluntarily go alongside their human trainers as partners, typically out of competitive spirit, having something new to do, general curiosity or fascination with humanity, or just because their trainers seemed like pretty cool dudes. Additionally, humans have a certain aura about them that helps Pokemon get stronger, faster compared to if they lived in the wild--and Pokemon seem to have a sense for this.
4. How violent are battles? Do they cause serious injury, or just a few light scratches?
Compared to what you'd see in the basic animations of the games and the largely bloodless carnage of the anime, more violent than canonically depicted. I'm not familiar with how the mangas do it, but perhaps more at that level? Cuts draw blood; punches leave bruises; heavy blows crush bones and probably give a lot of organ damage...
Buuuut Pokemon (and humans!) are very resilient in this world, both due to a significant environmental heal factor (PCs, Orans, Reviver Seeds, and so on) and natural Pokemon resilience. Across almost all canons, humans have been shown to have near-cartoon levels of physics as an in-universe property, and not just as a gag. I ran with that (After all, humans had to have survived living alongside Pokemon somehow...)
So, in the end, if we, Earthlings, were transported to the world of Pokemon, we would probably be downright horrified at how bloody these fights would be. But everyone else would look at us funny and say, "Okay, well, go back to your My Little Ponyta I guess." Because I think a good analogy for this is, the carnage seen in Kilo and the trainer world of my headcanon is to them the way we see how it's depicted in the anime.
Why is battling even a thing?
I brushed upon this in question 3, but battling is by and large the most common way for Pokemon to evolve to their strongest forms. This is inherently appealing to Pokemon for (actual) evolutionary purposes. Because Pokemon have evolved to battle for fun, and are extremely hardy folks, well... Battling just became part of the world culture.
As for Kilo, the same thing applies, even without humans to accelerate it: Battling is fun. Pokemon were built for it. In Kilo specifically, there are even scriptures in the Book of Mew (whether it's true or not) suggesting that Mew enjoyed battles because it was considered a "celebration of life."
Suppose that was a bit of a word wall, ahaha... Well. I hope that clarified a few things on my end of things.