Do dead position draws take into account the 50 and 75 move rules?

In the FIDE rules, a game is a draw by dead position if checkmate is impossible through any combination of moves. Declaring a draw by the 50 move rule is never explicitly said to be a move, and of course the 75 move rule is automatic. If a game was being drawn out to the latter limit, (say it was move 72/75 for instance) and mate/capture/pawn moves were all impossible within the 75-move limit, would that count as a draw automatically, with the game being position being considered dead because no mate could occur before the game would be forced to end? Similarly, if a player ran out of time OTB in a position where mating (and capturing and pawn movement) would not be possible within the 50 move threshold, could they rightfully claim the game was already drawn because of the fact that a 50-move claim is not itself a 'move', and no mate would be possible before such a claim could have been made? Or does the dead position rule completely ignore both the 50- and 75-move rules and allow infinite moves when determining if a position is dead?

If you're asking about here, there isn't a method to try and determine a dead position. You would have to reach either a triple repetition of position or 50 moves.

If you mean OTB, a truly dead position, one where there is no way to make any progress (locked pawns, no way to penetrate or sac material, etc), that is a draw, but if there's just a theoretical draw, because play will probably reach a 50 move rule, but mate is still possible by any series of legal moves, then the side with no time will lose if the side with time could possibly win (with worst play by the opponent).

"If a game was being drawn out to the latter limit, (say it was move 72/75 for instance) and mate/capture/pawn moves were all impossible within the 75-move limit, would that count as a draw automatically, with the game being position being considered dead because no mate could occur before the game would be forced to end?" ++ No.
https://handbook.fide.com/chapter/E012018 
'1.5 If the position is such that neither player can possibly checkmate the opponent’s king, the game is drawn (see Article 5.2.2).'
'5.2.2 The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’.
This immediately ends the game, provided that the move producing the position was in accordance with Article 3 and Articles 4.2 – 4.7.'
'neither player can possibly checkmate' and 'any series of legal moves' impose no limit on the length of the series.

"Similarly, if a player ran out of time OTB in a position where mating (and capturing and pawn movement) would not be possible within the 50 move threshold, could they rightfully claim the game was already drawn because of the fact that a 50-move claim is not itself a 'move', and no mate would be possible before such a claim could have been made?"
++ You have to claim a 50-moves draw when you have the move right after its condition has triggered or is about to trigger after your move. When your flag has fallen, you can no longer claim.

The 75-moves rule is automatic, but only if it has triggered before the flag has fallen.
'6.9 Except where one of Articles 5.1.1, 5.1.2, 5.2.1, 5.2.2, 5.2.3 applies, if a player does not complete the prescribed number of moves in the allotted time, the game is lost by that player. However, the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves.'

"Or does the dead position rule completely ignore both the 50- and 75-move rules and allow infinite moves when determining if a position is dead?"
++ Not infinite, but any finite series of legal moves suffices also if it exceeds 50 or 75 moves.

I got this,..

This is an excellent question and answer. The issue also applies to draws by repetition (3rep & 5Rep). Over the board, clearly 50M & 3Rep require a human decision, so DP could not apply. However 75M & 5Rep are automatic, so one might think that DP could see them. It comes down to the precise definition of what is a legal move. Moves which can never happen, due to DP, xRep or yM are "legal but unplayable".
In the chess problem world, we have the Codex to refine the FIDE Laws:
- 3Rep is automatic
- unless otherwise specified, 50M is automatic for "retro problems" but otherwise does not apply.
- 5Rep & 75M do not apply.
- unless otherwise specified, DP rule applies by default only to "retro problems".
There are non-retro DP problems which specifically engage the DP rule, where the task is to reach a dead position in a certain number of moves.
The current thinking is that for problems, the DP rule can see 3Rep & 50M for retro problems only. This seems to strike a good balance, and protects interesting classes of problems without being over-complex. I even hope this can be put in the Codex during 2024.