Diplomatic Immunity quest won’t start

You cannot get back there without console commands. The console command you are looking for is "coc " and you'll probably need to do it again to get out.

Use coc skuldafn02 to get there. The codes for Skuldafn are skuldafn01 trough skuldafn04. 02 should put your back in the Skuldafn temple. coc whiterun might bring you back home. Not sure about the last one.

The altar dragons actually respawn, it just takes some time. Perhaps do some fast traveling, or use Wait (but not at the altar). Many reports are that re-spawning takes 30 in-game days, but I've had locations take only 10.

I just tested Skyborn Altar. 10 days was sufficient for the dragon to respawn. I find it fastest to look at the date on the map screen at bottom right, add 10 for your target, and fast travel the longest distances to pass time. Markarth<->Riften is ~26 hours.

From my experience if you arrive too early you will have to wait all over again, better to load back before the arrival.