An authoritative guide to the DUP election by Northern Ireland politics expert Jonathan Tonge: https://theconversation.com/dup-leadership-election-qanda-all-you-need-to-know-as-edwin-poots-and-jeffrey-donaldson-160670
With such an exclusive election there could even be a tie, but there is no procedure to deal with that.
Nothing that Theresa May can offer the DUP will satisfy, reckons Northern Ireland specialist Jon Tonge: Even turning up on bended knee with another billion pounds won't do it
So that leaves three outcomes: trying to get a better deal from Brussels; a general election; or a second referendum. (One might add a fourth: a Brexiteer Conservative PM who goes for a 'Dunkirk spirit' no deal).
The DUP has not really done anything to enhance sympathy for Northern Ireland in England as they appear to be intransigent and backward looking. No doubt they would say they are protecting the interests of their constituents who continue to return them with (in most cases) big majorities. However, more vision might protect their interests better in the longer run.
Recall elections were a feature of Californian politics when I taught that, but they are new to British politics. The attempt to hold one in Northern Ireland is here examined by Jon Tonge who explains why Ian Paisley had a narrow escape: Preposterous secrecy
One of the least discussed aspects of 'snowmageddon' is that it could obscure any border between the two parts of Ireland, hard or soft.
Meanwhile, the issue threatens to become the most intractable in already difficult Brexit negotiations. Here are some reflections on whether a soft border is possible without a customs union (the answer is no): Hard issues
