Baron Nashor

No. Baron Nashor enters the Baron Pit directly — his entry location is replaced, not his position on the board. A move is defined as a permanent changing position from one space on the board to another.^[446.1] Baron Nashor is entering the board for the first time (a zone change), which is not a move.^[446.2]

A non-combat showdown opens during the resulting cleanup.

If an opposing unit arrives at the Baron Pit during the showdown, a combat is staged and the non-combat showdown becomes a combat showdown during the following cleanup.^{[316.8.b.1.a][323.9][323.14]}

If the showdown closes with only Baron Nashor's controller's units present at the Baron Pit, and they have not yet scored it this turn, they conquer it and score.^{[348.2.a][348.2.a.1]}

Yes. Winning by conquering at match point requires having scored every battlefield on the board that turn — including the Baron Pit.^[471.1.b.1] If it hasn't been scored, the winning conquer draws a card instead.

Winning by hold is unaffected, as it has no such requirement.^[471.1.a.1]

It depends on whether the Baron Pit is already on the board.

If the Baron Pit is not yet on the board, Baron Nashor enters the Baron Pit, not your base. As Baron Nashor is played, his "as you play me" instruction fires during the playing process — the Baron Pit battlefield token is added to the board.^{[135.2.b.3][187.9]} Under Mageseeker Warden's restriction, your base is the only valid playing location — where Baron Nashor is nominally played to — but his replacement effect overrides where he actually enters.^[355.2.a] His "If you do, I enter there" then functions as a replacement effect that applies as he enters the board, replacing his entry at his original play location with entry at the Baron Pit.^{[369.3][370.1.b]}

If the Baron Pit is already on the board, the "as you play me" instruction does nothing — the Baron Pit token is only added if it's not there already — so the "If you do" condition is not met. Baron Nashor simply enters his base, the only valid playing location under Mageseeker Warden's restriction.^[355.2.a]