Igo

No — placing a stone that would leave your new stone or chain with zero liberties is not allowed, except for the special case where that play simultaneously captures opposing stones and thereby gains liberties (so your stones remain); see page 1.

Stones of the same color connected orthogonally form a chain and share their liberties (diagonal connections do not count); if the whole chain has no liberties it is captured and removed; see pages 1 (sections on Liberties and Chains).

Yes — playing on the last liberty captures the single stone or entire chain, and the captured stones are taken off the board immediately; see page 1 (Capturing).

No — you are not allowed to play a move that recreates a previous position (immediate recapture that would repeat the position is forbidden) to prevent infinite repetition; see page 1 (Repeating positions / ko rule).

A player may pass instead of playing; whenever you pass you give an extra stone to the opponent to add to their captures, and when both players pass consecutively the game ends (white must pass last to equalize moves); see page 1 (When does the game end?).

Score is territory plus the number of stones you have captured (captured stones can also be removed from opponent territory); White receives an extra 7 points to compensate for going second; see page 1 (Object of the game).