Trying to add a hotmail to another hotmail

I have two options at this point:

Send email using your provider's server (recipient will not see your address)
Send email using's server (recipient may see your address)

My question is, if the recipient doesn't see my address, then what do they see?

If you have time, I also need to fill in another question. It is:

Outgoing (SMTP) server: what is it?