Re: The optics don't look good
In short, yes. If the government falls, it's no longer in charge of the Parliamentary timetable.
So there's 14 days to form a new government, in which time Johnson remains caretake PM. The Queen will pick someone to be PM who has a chance of commanding a majority. So if say Corbyn could come up with agreement from the Lib Dems, SNP and a hanful of others (including about 5 Conservatives I think) - he could be PM. After passing a subsequent vote of confidence.
That's not likely. So it would have to be a less partisan figure, I suspect. The problem being we don't have a tradition of caretaker governments in that way. But it could be done.
Also Johnson would get a chance to win back his Tory waverers and have a new vote of confidence. Or the Tory party could sack him and appoint someone else as leader, who could win a vote of confidence.
After 14 days of fun-and-games fails Johnson as caretaker PM advises the Queen on when to have an election. It's a minimum of about 4 weeks. The threat being to hold it after the 31st October. I really don't think that would wash, and I suspect the Queen would tell him to get stuffed. Given the subject of the election would be Brexit - that would be taking the piss too much. The Privy Council could be used as constitutional cover for this. I know there's been talk of it, but it's an obvious piss-take. It would probably lose him the election, and the EU could refuse to play ball by granting an extension until after the election date anyway.
Getting a new government doesn't get us very far though. The EU are not offering much in the way of options but sign the withdrawal agreement. Johnson has the best chance of renegotiating the backstop (and my betting is there's no little chance of anything non-cosmetic) at this late stage. I think a new government would struggle to get agreement even for an extension of membership - unless for something very specific that suited the EU too.
The one-sidedness of the backstop, and the EU negotiating teams own undermining of it by saying they could use it to force the UK to stay in the Customs Union indefinitely is the problem. There seems little compromise on offer on this so far. That makes the Withdrawal Agreement extremely unattractive - but it's the only game in town to avoid no deal. Maybe slightly modified? The alternative is to revoke Article 50 (there's unlikely to be time available for a referendum). I don't see Parliament daring to do that.