Is this a profitable play?

What are your thoughts and questions to that hand?
Fold on the Flop
Fold on the Turn
Fold on the River

And fold preflop of course. You've forgotten. Against a 3x open from MP at NL5 come on.
As played I wouldn't mind a check/raise on that flop though.

85's is the very very bottom to call this pre. I fold this.
Yup, good board for your check-raising range OTF.
From that point, I diagnose you the FPS syndrome.

I would say 85s is too wide preflop. OTF I prefer def xr, really good hand to do this with, bunch of cards you can continue on. Turn is fine imo, but you shouldn´t shove this R, ppl will station on this runout too much imo.

Preflop: Better to fold. I think 86s might be a defend, but still marginal.

Flop: Gutter + BDFD is better to call, but might also x/r. Folding is out of question. I personally lean towards x/c, because we don't have too many nuts here. 44 and 66 make 6 combos of sets. We could x/r those (call is good as well) and as bluffs use 75s (4 combos). If we add 85s, it's not that bad, we'd have 8 combos of bluffs then.

Turn: I'd probably play the same vs aggressive opponent. It's hard to call vs this sizing vs passive or unknown, we don't get a good price with 29%:

Board: 4♥6♠Q♣ 4♠
       Equity     Win     Tie
MP2    20.78%  20.78%   0.00% { 8s5s }
BB     79.22%  79.22%   0.00% { QQ+, 66, AQs, KQs, AQo, KQo }

River: is a spew. Not sure what better hands you are trying to fold out.