I would expect that both sides of the network would compute the next block with different difficulties, and when they reconverge,
1) Why just "both"? Consider a multi-way split.
Nothing changes about the mathmatics in a more than 2 way split, beyond the odds of this actually occuring.
Edit: even in the 2-way split the majority can get unlucky and minority lucky. The difficulty doesn't simply follow the majority after the split.
The two sids would have to be very well balanced for the minority to come out ahead, and such an even split is unlikely. Even if such an astronomically unlikely event were to occur, nothing about the analysis changes except which side is the majority.
2) Game theoretic issues are: when building next block do I trust myself or do I trust the blocks of unknown provenance after the split?
This is decided by the network protocol, and it doesnt much matter which you 'trust'
Whcich banch is going to be more profitable? Is there a profitable way of feeding "bait" blocks to the split-off portions of the network?
Sure, by double spending on both sides of the split. But splits can be detected, particularly if you are on the minority side, so wise vendors don't sell large ticket items during a network split.
3) Control theoretic issue is: reconvergence may take very long time, more than the default 6 dekaminutes and involve oscillations (flapping in the terminology of BGP route convergence).
You don't really understand what's happing here, do you? Do you even understand what you just wrote?