No such thing as approval of resignation notice. The day your boss acknowledges (i.e. sign and return to you) is the day you resigned, then count 1 month or whatever from that date.

Yes it should.
Keyword being should.
If push comes to shove, do you have any evidence that you delivered your letter to HR? Did they give you any dated acknowledgement?

Then that's fine. Just follow up using that email to HR and your boss and everyone using language that says look here guys i have resigned on XX date and my last day will therefore be YY date.

Having said that, if you don't want to burn bridges with your current boss you could consider following his proposed timeline for when your last day of service is.