checkmate with bishop and king
No; as long as those are the only pieces on the board, it can't be done. However, if your opponent has more pieces, than you might be able to try to trap the king against them in a corner and score the checkmate that way (in almost all cases though, it's not possible).
While we are on that topic, it is also impossible to checkmate with just a knight and a king, but if you have a knight and a bishop it IS possible to score the checkmate. If you happen to be interested, here is a link to a pretty good video that explains how to do it (it is definitely one of the most difficult checkmates possible if you ask me) https://www.youtube.com/watch?v=r3EqM17jvOc there are also places online where you can practice these mates if you so happen to also be interested in that. (just look up bishop/knight checkmate practice and you'll find one)
Simplest example. Black King on a8 and Pawn or Bishop or Knight on a7. White King on c7 and Bishop on b7.
For a King and Bishop to win on time (FIDE) the opponent must have at least a Knight and/or a Bishop (traveling the opposite color squares from the player's Bishop) and/or a Pawn (which could be promoted to a Knight). Otherwise it is only a draw if the opponent flags in FIDE. In USCF King an Bishop is only a draw when the opponent flags unless there is a forced mate (Black Ka8, Pawn a7, Pawn a6, Bishop b8 Knight c7, White Bd7, Kb1 and Black to move flags). In Chess.com it is only a draw when the opponent flags even when there is a forced mate.
Yes. white to mate in one moveNotice how the pawn is blocking the black king's escape. If it a king and bishop vs a lone king, the game is a draw.