Yes, を is grammatically correct, but usually omitted. When speaking, を is very commonly omitted from almost everything. The only verb that I can think of that grammatically shouldn't use を is かかる, as in 時間/円 かかる。
My guess is probably not, but because I learned my Japanese conversationally, I'm curious whether this is just commonly omitted and in fact grammatically required, it's grammatically incorrect to have a particle, or it doesn't matter either way.
Yep. I hear it too. I think TTS software treated 円払い as a single word (dunno, "yen payment" maybe?) and thought that rendaku occurred there, of course being totally wrong.
It is read as レストラン・で・に・せん・えん・はらいました。