All the smaller ones will feel like plastic keys. Not a piano.

They have to be plugged into a computer to hear anything. There are no built in sounds.

Linux would be mostly up to the manufacturer. But many are class compliant and I’m sure Linux is too. So would work as long as both of those are true on the controller you choose.

No idea about apps/software sorry

If they're controllers they will need a computer and most likely a software to produce sound, which is coming from the software Digital Audio Workstation or a Standalone software. On the other hand keyboards and other synth work stations will have a library of piano and other sound that won't need speakers or a computer since its all baked in, those are more expensive but will work on their own. If you want the piano feel you should look into weighted keyboards that will mimic the feeling. Those are the basic features, you can shop for what adjusts to your needs looking for these keywords on the specs sheet.

I currently have a Korg MicroKey Air, the 37 key variant. Some people don’t like it, but I’d recommend it as it has a pretty good build quality and feel for the price, and the Bluetooth is particularly good for convenience, along with its weight.

As others have said, keyboards like these unfortunately do not usually have audio come with them as controllers are almost exclusively meant for use with software. They aren’t usually used widely for playing as producing, so playing a midi controller will be naturally different than regular piano. That also being said with the lack of audio.

If this lack of audio is a problem, then honestly the lowest price sound-controller I would recommend would probably be the Yamaha Reface, which would run you double your budget. I do have pretty high standards when it comes to electric piano sounds, but I urge to say that keyboards under this price tend to sound… off.

Also, if you find a good model over your price, do not be afraid of used keyboards! As long as you have a good way of returning it and you clean it thoroughly prior to use, it is simply a good money saver.

So, TL;DR, if you are looking for one recommended model in the price range, I would maybe look into a Korg MicroKey Air. Hope you do well!