When splitting a 240v circuit, what happens to neutral's amperage?
Delta. stop Stop STOP!  Wiring a subpanel will not fix your problem.  Your problem is a GFCI trip, a subpanel won't help that! Fat wire and dedicated circuits won't help that! GFCI stands for Ground Fault Circuit Interruptor. You have a faulty appliance or possibly bad wiring that has a ground fault, the appliance is leaking to ground, or leaking through you and shocking you. If it's always one appliance, it's the appliance.  If any appliance causes it, look in the outlet wiring for a neutral touching ground.
If you still want to wire this subpanel, awesome. You have a serious code violation with bathroom receptacles sharing a circuit with non-bathroom receptacles and it'd be good to be rid of that, and more power in the garage is never bad.  But I want to be clear, when you finish your masterwork and plug the same tool in, it's gonna trip the new GFCI.

It depends how you wire your subpanel, but either way, you'll be fine.  North American panels have two poles or legs, as shown here.  So you are putting in a 40A subpanel, 4-wire (hot hot neutral ground) with #8 wire.  Suppose you fully load two 20A circuits, one at 20A the other at 18A.

If you put both 120V/20A breakers on the same pole (via a double-stuff breaker, or in two spaces both &quot;black&quot; on the linked illustration), the black hot will load to 38A, the neutral will load to 38A, and there will be no activity on the red wire.  You could load that to 40A@120V  also.

If you put them on opposite poles (via a 2-pole breaker or in 1 black and 1 red space), then you will evenly split the load.  The black wire will carry 20A, the red wire will carry 18A, and the difference of 2A will be carried on the neutral. Note that this is uses only half the capacity of your #8 40A wire. You can carry another 20A@240V.


To answer your questions:

Once your subpanel is in, you can wire 2 subcircuits with a shared neutral (aka Multi-Wire Branch Circuit or MWBC), but I don't recommend it, because it does not play well with either GFCI or AFCI.  This is largely a deprecated wiring method.

The other problem with this method is the danger of overloading neutral.  You must put both subcircuits on a 2-pole or handle-tied breaker for common trip and opposite poles, so neutral carries only difference current.  2-pole breakers are easily confused with double-stuff breakers, and if somebody &quot;tries to save space&quot; bymoving it to a double-stuff, it overloads neutral. /3 cable is the right way to wire the subpanel's feed, but stick with /2 for branches.

The best way is in my second example above, conscientiously putting 120V branch circuits on opposite poles to the extent possible. I would recommend using a size bump to #6 wire, because the cost differential is not large, and this bumps you to 60A capacity, enough so you won't be worrying about power for a good while.
Keep in mind you are allowed to oversubscribe a panel: if you had a 60A subpanel with twelve 20A 120V breakers evenly balanced on the panel, and also one 20A 240V, that is fine, because you are rather unlikely to max out all of them at once.  If you followed along you know we have 140A of breakers on each 60A leg.  That is fine.