2nd-layer apps where the additional functionality runs outside the core blockchain process.
Will app developers be able to update their original deployed code on the 2nd layer? In other words, will the app developers be able to change the rules of the game at any time and have users follow them automatically?
Yes. An upgrade to the rules of a decentralized 2nd layer app is exactly analogous to a hardfork in a 1st layer chain. If the users are ok with your rule changes, they will just continue to rely on your upgraded 2nd layer API servers, and they will all operate under the new rules.
On the other other hand, if they don't like your changes, they can launch their own 2nd layer API servers based off the old rules and rely on those servers. So you can potentially have a split fork, where some users use the new servers and other users use the old servers. The biggest difference to forking on a 1st layer blockchain is that these two 2nd layer forks can both operate seamlessly on the same 1st layer blockchain (so split forking is a little "cheaper" in some sense).