First name? I guess that's your call. I wouldn't go any further than that though being mindful of the board rules.
Well, if her answer was "sure" then I'd go. I mean why not? She sounds like she's into it.
If you received a "maybe" or "I'll get back to you" or something to that effect I'd leave it alone.
It's really important for you not to have any expectations about this. Take it for what it is right now. A night out.
It's a balancing act. If she isn't acting like you feel she should or maybe it doesn't go quite as good as you feel it should, and you react a certain way toward that. That would be a no no.
Take it for what it is and have a good time regardless. Be responsible for your own emotions.
Be confident, not needy or clingy and whatever you do, do NOT bring a your relationship or use one night out as some sort of guage.
There are many other situations where I wouldn't suggest you go but she sounds like she is open to it. So go... shine, show her a little something different.
As far as visiting the pups go? Your call. Are you comfortable with her being there alone? If so, go find something fun to do during that time and see how it goes with you not there. You definitely don't want to give the impression that you are standing around anxiously awaiting her arrival. Know what I mean?