I don't think you need to split it up for each individual discipline. You just need to understand how big of an edge you have in each area and how high of variance each discipline will be.

MTTs you generally want 100-200 buy ins for whatever stake your playing depending on your edge. So if your playing online and your total bankroll is $1,000 you should be playing $10 mtts. But the $10 mtts are really soft and you have a big edge so you its pretty reasonable to play them with only $500. Now lets say instead you have $10,000 and you're playing $100 mtts. Those are a lot tougher and variance is a lot higher, so maybe you need a total bankroll of 20,000 to be playing those games. If you play a live MTT and its a $500 entry, your edge will be much larger in that than a $100 online mtt. So its reasonable to play that $500 mtt with only 10,000 (20 buy ins) because your edge is so big and you can play smaller buy in stuff online and cash games to build your roll back up. Likewise, you may only need 15-20 buy ins to play a live cash game because the player are weak. If you're playing 1/2 with a $200 buy in you may only need $3000 to be rolled properly for that game cause your edge is huge. But if you were playing online cash you might need 50+ buy ins because your edge is smaller.

So I would keep the roll together and just look across everything you play. If you have $5000 roll then maybe you play 1/2 or 1/3 live cash games. $100-$300 live mtts. .25/.50 and maybe .50/1.00 online cash games and $5-$50 online mtts.

Thats just a rough baseline. If you're not a winning player at $50 mtts or .5/1 online cash then you shouldnt play them just because you have a roll to play them. Also, its important that if your roll gets smaller you adjust all the games your playing to be properly rolled to play them.

I would not include money that has to be paid towards bills and living expenses as part of your roll. Realistically, that money is not available for poker because its needed to be used elsewhere. Ideally you'll have at least a couple months worth of living expenses set aside so that you dont have to take money out of your roll to go towards living expenses when you go through a downswing.

Not exactly sure what you're asking. But if you're not comfortable playing at a higher stake than you're already playing you don't need to move up stakes just because your bankroll is now big enough.
For example in live cash games you might have a bankroll that lets you play 5/10 nl. You play and do well and your bankroll is now big enough to play 10/20 nl but that game is tougher and has a lot of good players in it. You don't ever have to move up to the game. Also, you might be more profitable at the 5/10 game if it has weaker players or if your skill level is not high enough to play the 10/20 game yet.

Yeah if youre over rolled for the games your playing I would split it up and put some in investments/savings. As long as you have reasonably quick access to it if your bankroll starts to take a hit.

I play mostly mid with some mix of high stakes mtts online and the occasional live mtt. I keep a decent chunk of bankroll just available for buying in to mtts but then put the rest into various investments that could be liquidated back into cash reasonably quickly if i needed access to it.

My avg buy in is somewhere around $65. I keep about 200 buy ins available to be used as entry fees with the more in various investments/accounts I can liquidate and transfer either to bank accounts or poker sites within a week if needed. When I win a decent chunk in a tourney I withdraw and put a certain amount in my bank account for living expenses and the rest into other investments.