1, It fits with the psychopath, because his intent is to continue forcing himself on her till he no longer enjoys it. The footage ensures she stays complacent while raping her.
2, the "Secret Gallery" isn't something that is in the games lore. It is the $100 tier photos he adds to the game at each update.
3, At the start of the game he willingly and without hesitation gives MC money, till he says MC should just do work to earn money instead.
4, You describe the MC. He doesn't think he can fail raping his teacher, but does.
And not all psychopaths kills. It comes down to "what would cause a greater problem". The husband knowing that the kid was at the apartment could finger him as the killer and he would be sent to jail. A psychopath would know this and work around it. The blackmail and recording is a much better option since it is part of "his game". He is in control with it.
