Am I able to apply for a costco credit card on sundays?

Yes.

You should, I did on a Saturday, but I'll warn you right now, they have a cash back card, but it pays out yearly and my understanding is that it's mainly for Costco. But i may be wrong. I like it, especially the quickness of paying it If cash back is main factor, i am happy with the Capital one Quicksilver, has cash back and multiple pay out options.

Yes at the Customer service desk.

Yes, its completely possible, u can direct contact with CS @ Costco