How can you give some checkmate troubles to black with move no.1?
I got some chess books Y'all

No, definitely 3.Qb2+ 3.Qc3+ e1Q 4.Qxe1 Nd2 and the queen is out of the game, I don't see any hope for stalemate.

1.Qe5 Qa5+ 2.Kxa5 Kb7 3.Qb2+ Ka7 (3...Kc8 Ka6, and white threatens Qh8#, if 4...Kd8 5.Qf6+ Ke8 6.Qf7+)(3...Ka8 4.Ka6 1-0) Then what?
Qxf1 ?