They don't ever NEED to but it can be useful as people have said :)
Babies often don't like to be hauled in and out of contraptions all the time, so if you're going to be in and out of the car (e.g. driving to several shops/appointments) then it's easier for them if you can just keep them strapped into one thing rather than moving them around constantly. Plus if they are asleep like you say, it can be less disturbing and less likely to wake them up.
That said, do be aware of the positional asphyxiation risk - the position in a car seat is quite upright to allow for crash safety, and this can affext a baby's oxygen saturation levels over time so it's recommended to take them out and give them at least a short break, once every 30 minutes for babies up to 4 weeks old, and once every 2 hours for babies older than this.
You can buy car seats which have inserts to put the baby in a flatter position, and even car seats which have a flatter lying position due to a recline (often marketed as "lie flat"), although the advice doesn't change because there's no research on these seats specifically, it may help you feel more OK with relaxing these guidelines. It's just advice so it's your call to make. Personally I've always felt OK letting my babies finish a nap they started in a car seat, as long as it didn't push them far over the guidance, but I wouldn't put them in a car seat specifically to nap or sleep in the house, as I feel this is an unnecessary risk.
Also in general because car seats are fairly curved in shape (even the lie flat ones) and the baby is restricted in them due to the padding, harness and inserts it's not a good idea to keep them in the car seat for too long in general or use the car seat as your main carrycot/pushchair, they do need some time to lie flat and stretch every day for their development, learning to roll, push up, sit etc :) but I think this is fairly obvious, it's the time limit advice that isn't quite so clear.
As far as which car seats fit which pushchairs. There is one type of adapter, known as maxi cosi adapters, multi-brand adapters or "universal" car seat adapters which many brands have designed their car seats to fit on and which many popular pushchair brands produce for their models. These pushchairs usually list a small number of the seats which fit on them, but they are often out of date and/or incomplete as they really want you to buy their own branded car seat as more £££ for them. The easy way to spot one of these prams is that they will say their pushchair is compatible with something by Maxi Cosi or Cybex. But the full list is as follows:
All infant seats from:
Maxi cosi
Cybex / GB
Nuna
Besafe
Recaro (nb, the two combination seats use an interim adapter)
Avionaut (there is one branded as Venicci Ultralite in UK - no other Venicci seats fit)
Then also:
All Joie infant seats, except Juva
Cosatto Dock
Axkid Modukid Infant (with interim adapter)
Britax car seats use a different type of adapter, but also fit with a large range of pushchair brands. Generally you find that brands which are sold in the US or Scandanavia have Britax adapters available, whereas brands which are unique to the UK or mainland Europe often don't.
If you can only purchase a set as a full travel system/3-in-1 and there's no option to change the seat to e.g. a Maxi Cosi, it's likely they only accept their own brand car seats. And Silver Cross, Graco, Chicco, Hauck, Ickle Bubba etc car seats will only go on the matching brand's prams.
So it's not always the case you need to choose one before the other.