Re: phrasal verb
It's not synonymous with dating, but it means to ask to go on ONE date.
You should ask her out some time = You should ask her out for a date.
If you are dating someone, you have a relationship, you are a couple.
If you say "I've been asking her out" it sounds as though you have repeatedly asked for to go out on a date with you (and she has been refusing).
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.