If you want not shitty speakers and subwoofer, you're going to need to up your budget by a lot.

The Edifier R1280DB is within your budget and have Bluetooth, but no sub out.

The Edifier R1280T has a sub out, but no Bluetooth.

The Klipsch Promedia 2.1 is the best sounding 2.1 system under $300 IMO. No Bluetooth though. They used to have a Bluetooth model, but I don't think it worked very well.

Here's a new offering from Klipsch that looks promising. It has Bluetooth, but is a bit above your budget.

Otherwise, look up something from Kanto or Monoprice if you're willing to up your budget.

If i were to buy this amp and pair it up with the pair of these headphones, will it even work or will it have like no or reduced sound or something?

That would work just fine, why do you think the sound would be reduced? If anything the opposite would be the case because amps, well, amplify.

also, can i alternate between using the amp and using the speakers i plan to buy as well?

Sure.

Can i still stop the distorting from my headphones with the amp or do i need to just buy a pair with more ohms?

That depends on what's causing the distortion. But if it happens with very loud music it might just be the headphones themselves.

Also, don't do that. It's a terrible idea, don't destroy your ears, you'll kinda need them for the rest of your life.