Yes, this is why Kamala looks to be the best alternative. She is part of his campaign so if he was to step down she would inherit all the fundraising their campaign has done so far.

From my understanding, yes she would be sworn in and function until the next candidate was elected.

As she is the current VP candidate, I believe she would just get bumped up to the presidential candidate and they'd need to decide a new VP.

Now what I am not quite sure of is what would happen if he died mid-election after people have already voted for him. Would they do another run of ballots? It ultimately wouldn't matter much, but I would be curious what the mechanics would be.