Because he's a RFA, Burakovsky cannot make any less than he made last season. He has to make at least $3.25M on his contract.

Not necessarily, if he wants to stay in Washington they could do the same as the Caps did with Smith-Pelly or the Jets did with Morrow, decline the QO but then negotiate something for less to fit the player in under the salary cap. If Burakovsky was to do that with the Caps it'd obviously be a one year "prove it" deal but I could see it happening if he wants to stay with the Caps, if he just wants to chase the money then something else will happen.

If I'm not mistaken he only has to make that much if the team gives him a qualifying offer. If they don't he becomes a free agent on July 1 and can be signed by any team (including the Caps) for any amount.

This is correct. I wonder how often a player signs with the same team without a qualifying offer though...

It's not common at all. And if I'm Burakovsky, I wouldn't want to willingly take less than I am required to get via QO. Take a look at Mattias Janmark in Dallas. Not worth his contract last season, but Dallas gave him a similar 1-year "prove it" kind of contract this season, and it was the exact same as his contract last year.