Lots of questions. Thanks for taking the time to actually formulate things well though, even though I'm certain you won't quite get as nicely formulated answers.
I'm really tempted to say that you should just pick a program that has those in it and use that for awhile. I think one of the Bill Starr 5x5 has those, and/or maybe Starting Strength, but I'm not sure.
I don't know. You can get benefits from all of them. I will say certain isolation exercises are known to be worse for the joints than the others. Heavy skullcrushers is a good example of a problematic exercise.
Yes. In the context of strength training, isolation work is used to bring up the weak areas so that the compound movements stress everything more equally. For instance, with bench pressing, some people might have very strong chests and front delts -- they're able to get the bar 90% up with just those -- but are weak at locking it out at the top. In this case, doing some isolation work for the triceps allows them to progress on the bench.
They both have their place.
This is one of the reasons I think sticking with an existing program would do you some good for now.
This was tricky when I first started out too. What I suggest is that you try to take a short rest and see how you perform on the next set. If you take too short a rest, your body won't be recovered enough and you won't do well on that set. Eventually you'll learn what your body feels like when it's ready for the next set. For some sets, you only need a short rest; for others, a longer break... but basically you just learn when your body is ready.
Since it sounds like athletic performance is really what you're after, I'd prioritize that. As in... if the strength training is taking too much of a toll on you, then you need to adjust your [strength] program.
If you need/want to use cardio and/or plyo to improve your athletic performance, that should be fine.
However, make sure you increase your calories and protein. Strength training requires quite a bit of both in order to grow... then, cardio and other athletics will require even more. So you need to make sure your food is in line for the increased caloric expenditure. And, since muscle is built with protein, you need to make sure you have enough to build new muscle after your lift.
Some light stretching or mobility work... things like arm swings, leg swings, stuff like that... and maybe some light cardio like walking on the treadmill for a couple minutes, just to get the joints and muscles warmed up is always good. I don't always do it, but it's good to do.
I've lifted where I've just jumped directly into work sets without using a lighter weight. I've also used a ramping approach starting light and working my way up.
I prefer the ramping approach personally.
This is where a good program will give you some guidance
First, core is more than your abs. Second, heavy squats and deadlifts, done with good form, will improve your core strength quite a bit. Your core is basically bridging your upper and lower body and holding them in place, so it gets worked hard.
But, I mean, if you want to do more, I'd suggest doing it after your primary workout. Two suggestions: ab rollouts, and heavy overhead holds.