Yes there are constitutional issue's around the Prime Ministers power to ask the queen to dissolve parliament, something which was solved with the now repealed fixed term parliament act. However I think if Boris asked the queen to dissolve parliament or call an election she would have to do so. Two separate things thought that usually intertwine, it's all very messy.
I think Boris is playing a dangerous game sacking Gove, however I think he had little alternative. I think if Boris is backed into a corner (which is the right thing to do by the way not supporting him) then I think he will call an election as I imagine his thinking would be along the lines of, I'm going either way but there is a small chance if I call an election. I think if Boris called an election all of a sudden you'd see the Tory party and media rally around him, they simply cannot be risking changing leader during an election campaign. I would not be surprised if an election is triggered at some point within the next few weeks. Boris is unlikely to survive it but he has that small chance, I also don't think Boris thinks he would loose an election.
like I said earlier two paths: 1922 committee kicks him out or the public kick him out (and hopefully he looses his seat too, let's put him out of all public office).