1) How do you determine if this is a client contacting you (via email) vs another provider? Does this effect how you answer the email anyways?

I don't know if you can really tell. I will answer honestly, because I don't have anything to hide with my pricing, etc. As long as the email seems genuine, and not spammy or scammy.

2) When someone emails you and just asks price and number of spaces (as this person did which made me wonder) how do you suggest a phone conversation to discuss the details or do you just respond via email?
If the email was literally just "how much and how many", I would respond with questions of my own. What age are the children, what hours are they looking for, etc. No response to that, and it probably was another provider being snoopy, if you ask me.
I stopped advertising my phone number as well. I would prefer now to have a couple of email communications first. I keep an open mind, because as we know, you lose a lot in terms of tonality, etc electronically. But it does help weed out some of the crazies. (ie: I expect you to watch my child from 6-6, rock them to sleep, provide a raw-food diet and NEVER EVER tell me that it was my child's fault. LoL)

3) Do they still get charged the b&a fee as I am open even if they opt to not have their kid with me for the full day or at all during that time?
I am not currently doing B/A, and I know that the "norm" is pretty varied. I charge a flat monthly fee for B/A that covers the daily. On top of that I have an extra charge for full day care (basically just covers the additional food). Admittedly, I do not "include" the full days, because I don't mind at all if parents make alternate arrangements for those days. In the summer, parents pay the regular full time fee.