While I can’t tell you what to do I can tell you what works for me …
I have recently upgraded to a kickr 1.5 with a kickr climb and I do enjoy it quite a lot
I also use a 2012 Marin Palisades Trail mountain bike on it and it works great… The only thing I’ve done was to change to a 25-11 cassette for tighter gear ratios. I found the stock set up’s gear spacing too much of a jump between gears while ‘road riding’
And since I went new cassette I also chose to get a new chain and chain rings so they’ll all wear evenly
So yes, a mountain bike works, and I’m happy with it
Other good stationary bike options to consider are from Wahoo, Stages, and Tacx (Garmin). They’re all pricey.
The mountain bike is almost certainly compatible with a trainer, but depending on the bike’s gearing and your power, you may not get sufficient resistance for a good experience. This mostly affects people using mountainbikes with a 1X drivetrain and a small front chainring (often 32 or 34 teeth). There are some workarounds for that but it’s not ideal.
You might consider getting a direct drive smart trainer (Wahoo, Elite, Tacx, Saris are common choices) and try it with the mountain bike. If you don’t have a good experience with it, or the workarounds for low gearing are too annoying, get a road bike. A used 8-speed bike would be the place to start if you want to keep the price down.
I would maybe go for the Core first, then later you can always upgrade to a stationary bike (I did the same once I knew the investment was worth it for me).
If you’re only going to use Zwift then go for the Kickr Core Zwift One, a much easier setup, and as you have limited space you can put it all away when needed, something which is harder to do with a stationary bike.
It looks like the Carrera Vengeance 24-speed bike comes with a 12-32 8-speed cassette, so that would be a safe choice if you want an additional cassette for the trainer. 12-28 would also work fine. The easiest gear on that would not be quite as easy as the one on the rear wheel now.
The kickr bike is actually quite compact compared to a real bike connected to a trainer and still having a front wheel in place. In the room where your front wheel might have gone you can place a 450mm industrial fan.
It’s also very practical in that you can adjust the size/fit easily. This doesn’t seem important at first but trust me it is when you have a serious crash and mobility issues.
Unfortunately the smart trainer bikes are expensive.
Kickr Core Zwift One comes with a single cog pre-installed, you change gear virtually in Zwift using the Zwift click that comes with the trainer. I believe it’s compatible with 8-speed bikes. If you go for this option you won’t need to buy a cassette. And you get a year of Zwift included.
But, the Zwift cog, click and virtual gearing is only compatible with Zwift. So if you’re planning on using other cycling apps then you’ll need a cassette for the trainer.
If you bought a trainer with no cassette installed then yes you would need a cassette and a Shimano lockring tool. Or just take the bike and trainer to a bike shop and have them sort it out. It may need minor adjustments to shifting when it’s put on the trainer.