First, I am so sorry you have to deal with this especially after a long day at work.
also, NO its not wrong to occasionally spank your child and it seemed like you took the appropriate steps when doing it (explain to the child why you did it and then comfort them after they understand why it happened).
Second, have you tried punishing them and following through with it? Catch them goofing off you warn them and then the second time its time out. Standing or sitting in the corner will show them you mean business but you NEED to stick to your guns. Once they see weakness they will run all over you. Do they do this with dad? have you guys spoken about this? You two need to be on the same page as far as bedtimes, keeping their schedules, and discipline.
third, i wouldnt discourage DD1's curiosity but I WOULD put a stop to her questioning you. Eventually explaining your actions will get tiresome and you will resort to "because i said so". lol. you should tell her that people think its rude to do that. she is a little girl and even though she might be smart she should NEVER be allowed to talk to anyone that way.
I understand its hard to take something away from one while the other is good but if you took chucky cheese from both of them then the one who deserved it will eventually get really mad at the other for missing out. Peer pressure could work to your advantage. haha.
Anyway, its not easy, my mom was a single mother with me and my brother. we were latch key kids in the afternoon but she was always home to make dinner and help with homework. women in tougher situations have done it so dont lose hope.