You have a lot of questions, but to answer a few:
On the fly re-calculation of the trajectory isn't the difficult, the Saturn V had this capacity in the mid 60s, Apollo 6 (an unmanned test flight) lost 2 of it's 5 engines on stage 2, and was able to recalculate it's burn times on the fly.
Broadly speaking, losing an engine and having to run your other engines longer is fine(assuming you aren't running with engines below their peak capacity and are able to just throttle up), as long as two very important issues aren't a problem.
You're not experiencing noticeable drag from the atmosphere. ie the earlier you lose the engine, the less effective longer burns are to make up for it and the more relevant a throttle up capacity is. This is because for the first portion of the launch, you're not increasing your rockets relative velocity, you're just trying as hard and as fast as you can to get out of the atmosphere. At a high altitude(as in tens and hundreds of km up), running a little longer means running at <1% of normal drag, so your spare fuel will be fine (max Q tends to be in the first 5 minutes of launch). But if you lose your engine at a point closer to launch, you need to spend much longer running your remaining engines to get above the atmosphere. Every second you are in the atmosphere, you're losing that hard earned velocity to atmospheric drag.
Longer burns than planned cause issues with your design, especially with reliability(which was NASA's worry) and for reusable systems with maintenance scheduling, since engines are essentially built around only needing to operate for a finite period of time. Frankly, compared to 99% of equipment designed by engineers, Rocket engines only need to last a laughably short period (even with testing). But then when you add in the 1st stage re-usability and a single engine out might mean that all the other engines now might have 20% higher operating time, meaning they need a heavier rebuild between launches. It might not be an issue on a single launch, but over long periods of time it's an issue if an engine had to run 20% longer 3 times out of 10. Remember with 9 engines, one engine out effects every other engine's operating time.
There are other things which complicate matters, ISP and engine effectiveness is effected by altitude as well(usually staging nozzle/bell designs take this into account). I'm not a rocket scientist (I've only designed a few 3d printable engine designs in HYSYS and solidworks for fun) though, so take my knowledge with a grain of salt. There are probably real SpaceX People here laughing at my simplicity, but they aren't allowed to post.