My 2 cents:
- What are the pros and cons of different propulsion types?
Treads are the slowest non-overweight propulsion, with normal treads clocking in at 160 and the fastest prototype treads clocking in at 135. They make up for this by providing a huge amount of support, have integrity and coverage enough that they work as armour on their own, and have bonuses to reducing recoil. For combat, treads require the lowest amount of slots to run without going overweight, although the benefit from this is not as good as it sounds due to their 100% trap trigger chance. They're fairly easy to acquire, sentries being a great source of treads. I'm not a fan of treads because of their slow speed, which is punished more and more later into the game, and the fact that although coverage and integrity is the best way to mitigate damage in early and midgame, resistances and evasion are the best ways to do this in lategame. Treads also allow you to literally crush your enemies by rolling over them, which is insanely cathartic if you've lost as many runs to Wastes as I have.
Legs are of moderate speed when not overweight, with most legs in the vicinity of 120, plus or minus 10, although there are outliers. They have decent support, usually forcing you to run 4 or 5 propulsion slots to avoid being overweight. Their main benefit is their ubiquity, meaning that you can very easily get basic legs from both grunts and hunters. I don't have a huge amount to say about legs beyond that - I find 5 prop slots wasteful when you could get similar speeds and efficacy at less energy cost running 2 or 3 with wheels. Tripod is a unique playstyle attached to legs, though, and I enjoy that quite a lot. Running 3 legs overweight means your speed tends to hover at about 150, 10 faster than treads, but you only have to use 3 propulsion slots, and you still get the other benefits of abundant sources and decent integrity/coverage, plus it synergizes pretty well with certain items scattered throughout the midgame and the lategame. Kicking is a unique benefit to legs, as well, which makes ramming much less likely (or if running 5, impossible) to cause self-damage. I haven't found a good use for it, but I suppose it is nice in a panic situation.
Wheels are the fastest ground-based propulsion when not overweight, at 80-100, but very low support means you'll probably never see that speed outside of the scrapyard. They suffer from very low integrity as well, meaning that stacking wheels to achieve that speed is a bad idea. The trick up their sleeve, though, is that they have extremely low overweight penalty. You can run several times overweight on wheels and match speeds with legs. Basic wheels are also very available as well, since most non-combat bots use wheels, and there aren't very many types of wheels, so you can run on these basic crappy wheels until lategame, though they do mean a noticeable difference in speed. I personally massively enjoy running 2-prop wheel builds, since they allow basic leg speeds going into endgame, support a very heavy tank build, and allow many utility slots to be evolved which is an incredible power boost. The downsides of this are that wheels need to be heavily armoured to be effective, and the early/midgame is never quite comfortable, since the best damage mitigation comes in later.
Hover, I don't really know a lot about. It's kind of a hybrid between traditional combat and flight styles. It's very fast and has decent evasion and support, but non-combat hover units have low integrity, and there's a huge power difference between basic hover units and combat/prototype hover units, kind of the opposite of wheels. This means that hover is really hard to get your hands on, and it doesn't allow for a lot of mistakes when it comes to losing propulsion. I can imagine that a build combining cooled and combat hover would be very powerful, but it requires a lot of setup, and I'm imagining usually a transition build in between would work? I don't know. If I'm specializing in combat, I'd rather go with wheels than hover, and if I'm specializing in stealth, I'd rather go with flight than hover.
Flight is the prime choice for stealth builds, and that's for a good reason. Incredibly high speed, low support, low integrity, and the ability to hop over other robots all combine to make it the best propulsion out there for avoiding fighting. Flight builds need to be very specialized, and usually have a transition build of treaded or legged combat in early game to set up. Once you get a handle on them, though, they are, in my opinion, the most reliable way to get a basic win. Flight builds suffer from being very fragile in early mid-game (-7, -6), but are more resilient than almost any other build in lategame, and they completely sidestep dealing with alert. However, this all comes at a huge cost - they cannot fight. Yes, I know flying brick exists, but for 99% of the mid and late game you still aren't fighting. But, if your goal does not involve combat at all, you'd do well with flight.
- How does your choice of propulsion affect your playstyle?
As I've stated, I think treads, legs, and wheels are the best choice for combat playstyles. Flight is the best choice for stealthy playstyles. Hover sits somewhere in between - you're fast enough to outrun anything that isn't flying or hovering, which means you can take on more of an evasive combat playstyle? I don't know. All other combat playstyles tend to approach similarity in endgame regarding speed and utility style. Treads do seem a better choice for kinetic builds because of the recoil reduction, but crit builds, the most effective kinetic build, don't tend to have a lot of recoil.
- How many propulsion slots should you evolve?
Depends on the build type you're going for. Treads should work towards 4 propulsion slots (because of 2-slot treads all but eclipsing 1-slot treads by lategame), legs should specialize in tripod or run 4-5 slots, wheels should run 2, flight should run 4-5 if hacker or aim for 8-9 by -1, evolving more slots as needed if flying brick or other heavy style. I don't know about hover.
- What is your favorite type of propulsion and why?
Wheels, definitely. Being able to achieve decent speeds with 2 propulsion slots means you can spend pretty much everything else on utility slots, and seeing as utility slots are the most powerful slot type, this means your resulting lategame build can be very strong. Downsides are that you have a loose mid-game, when good armour hasn't really entered the picture yet, and that if you have sloppy enough play to the point of build collapse, rebuilding is very difficult, as you don't have the ability to run prop armour (or easily transition to another build in lategame), and using your utility slots properly means you have a great number of specialized utilities that may be hard to replace.