I've done the Proving Grounds on a pure F2P acct with the following results:
Proving Yourself: Endless Damage (Wave 30) - You get the title "the Proven Assailant" added to your account. But, if you apply the title to your char, it disappears after logout. When logging in on any Remix char, the title is available again after various in-game actions are performed (like getting on a mount). Title has a level requirement (level 50?), so it is not available for use on any non-Remix character on F2P accts.
These achievements and titles were pretty hard on retail, so get them while Remix is up. The titles would be available for use if you do decide to sub and level your chars.
You're Doing it Wrong and
You're Really Doing It Wrong achievements worked for me (I did Endless Heals with a lv 20 Shadow Priest).
Proving Yourself: Endless Tank (Wave 30) did not work for me despite going to wave 60. Tried on a druid tank and a survival hunter.
Only got the
Proving Yourself: Endless Tank (Wave 10) achievement.
Same for
Proving Yourself: Endless Healer (Wave 30). I did get achievements for Wave 10 and 20, but not for Wave 30 despite completing 40+ waves on multiple classes.