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.^[441.1] Baron Nashor is entering the board for the first time (a zone change), which is not a move.^[441.2]

A non-combat showdown begins. The Baron Pit is newly created and has no controller when Baron Nashor enters it. Being played to a battlefield applies contested status to it, provided the entering unit's controller does not already control that battlefield.^[187.3.a.1] Because the Baron Pit is uncontrolled at that moment, a non-combat showdown is staged during the resulting cleanup — the same mechanism that triggers when Sneaky Deckhand is played to an open battlefield.^{[316.5.b][323.8][344.2]}

If an opposing unit arrives at the Baron Pit during the showdown the non-combat showdown becomes a combat showdown.^{[316.5.b.1.a]}

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.^[466.1.b.2] If it hasn't been scored, the winning conquer draws a card instead.

Winning by hold is unaffected, as it has no such requirement.^[466.1.b.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][184.9]} 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 (per Riot ruling) that intercedes when he would enter the board, replacing his entry to his base with entry to the Baron Pit.^{[369][370.1.a]}

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]