I can answer the question about which embassy is best for this. This really depends on where you're going in Schengen countries. According to this AXA and what I have experienced as well. You need to apply for the country that is your main destination. If none of the countries are your main destination (maybe you're going there for the same length in each country) then you need to apply for the country where you enter first.
For example, if you are going to Germany (4 days) and Austria (5 days) then you need to apply for Austrian embassy. If you're going to Germany for 4 days and Austria for 4 days then you need to apply for German embassy. I'm guessing that the invitation is just to proof that you're staying with your friend so you don't have to provide and hotel confirmation. But if your friend is willing to sponsor you for any expenses, then your friend needs to send you bank statement and passport as well. However, the length and how many entries really depend on the decision of the embassy, which you cannot guarantee.
But as I have done this before for my friend. you're likely to get multiple entries if you're invited by a resident in Schengen country for the maximum of 60 days (According to my friend's latest visa not from any website). However, the duration can be various, you might get to stay in Schengen area for example, 20 days then you have to go out and in again.
The answer is based on my experience.