No to both questions. It would be forced onto the UK anyway. And there's more to it than EU treaty changes. The backstop section in May's deal would basically subject:
Northern Ireland to new legislation passed in the EU, owing to the fact that Northern Ireland would stay aligned with the EU's internal market for e.g. food standards, etc.
The UK to new trade treaties passed by the EU and whatever else affects the customs union, owing to the fact that the UK in its entirety would stay in the customs union.
In both cases the UK (and NI) would have no say on paper. (In practice I can't imagine the EU not keeping the UK in the loop behind the scenes.)
That being said, several other points in case you're asking while wondering what to make of the backstop:
The reason there's a backstop to begin with is because of the Good Friday agreement. Had there not been a peace treaty that says there cannot be a hard border between Ireland and Northern Ireland, the backstop wouldn't have been necessary.
What the EU wanted originally was to only keep Northern Ireland in sync with the EU, in order to maintain an open border between NI and the ROI. Until late last year the EU had a red line whereby making the backstop apply to the UK in full was out of the question. May got a huge concession when the EU reversed its red line, and this led to the current deal.
The reason the EU is so dead set about the backstop is because not doing so would create an existential threat to the EU. The EU was created on the ashes of Europe after WW2, to perpetuate and defend peace across the continent. If it doesn't stand firm on defending the Good Friday agreement that Ireland (an EU member) passed with the UK, the EU would basically be sending a message that a) the EU doesn't stand behind its members and b) it's fine to fudge with peace treaties that affect EU members.
The backstop is a last resort thing. It's not supposed to come into effect unless negotiations break down completely or aren't completed by a certain date.
Also, if memory serves me well, there's some language in the treaty that gives the island of Great Britain a unilateral opt out -- so long as Northern Ireland remains in the backstop for as long as necessary. Or maybe there were only talks about it -- but if so I can't imagine why the EU would have rejected that. (I can't find the exact source or podcast where I came across this. A helpful reader will hopefully add a link in the comments or edit it into this answer.)
You're objecting in a comment that this doesn't actually answer the question, so let me make another point more explicit: the Irish backstop is completely unrelated to the UK's ability to influence treaties once it leaves.
The moment the UK is no longer part of the EU, is the moment where it no longer has any say on what's going on in the EU or on EU treaties. Period. Just like Norway or Switzerland, which align with parts of EU laws, without having any say on what goes on in the EU. (Not officially anyways. One might easily imagine some backchannel communications as a friendly courtesy.)