TL;DR DANGER! Something's seriously wired wrong
None of this fits with a problem in the hood itself. It all points to a problem elsewhere. I can't think of an exact scenario, but possibilities include a miswired MWBC, a loose connection that heats up under high load making other stuff work better, something wired in series instead of in a tree configuration, etc.
I would start with the hood connection. Disconnect it and check voltage hot-to-neutral, hot-to-ground, neutral-to-ground. I know you said "no ground", but many houses of that vintage (like mine) actually have ground in a lot of places just no (or few) 3-wire receptacles. If you have a metal cable and/or metal boxes they will often be ground. You're looking for 120V (+/- a few is OK, consistency is what matters, will treat as 120V for the rest of this) hot-neutral and (if you have ground) hot-ground and 0V neutral-ground. Check it (carefully) with lights, toaster, etc. off, with lights on, with toaster on. Report any change in the voltage as that is an important clue.
The next step is to check every junction box on this circuit. Which may not be so easy, so start with the easiest - the known kitchen and garage receptacles. In addition to checking for loose connections (and redoing/tightening them) and backstabs (move to screws) and replacing any receptacles that appear burnt up or broken in any way, note if there are any locations with extra wires - e.g., black/red/white instead of just black/white, as that would be an indication of an MWBC. An MWBC miswired can cause all sorts of problems.
Probably unrelated, but extremely important for safety: GFCI. Even if you do not have ground available (and you may have it, even if you don't think you do), you can and should install GFCI to protect kitchen and garage receptacles. Especially kitchen - the risk around water is simply too high to ignore when GFCI provides a huge increase in safety at a very low price. But that requires understanding Line vs. Load, so pictures and details will be helpful before attacking that part of the problem. In addition to safety, GFCI can often help narrow down certain other types of wiring problems because it will only work if many (not all) aspects of the circuit are properly wired.
Another thing to (carefully) try:
Take an extension cord, chop the socket (female) end off, wire it up to the hood and plug it in to a regular receptacle (GFCI or not doesn't matter, that's not the issue) that is on a different circuit. If that works (lights, fan) normally, plug it into the same circuit (i.e., where the toaster plugs in) and see if that works normally.
If it works on a different circuit, then the problem is in the wiring of the bad circuit. If it works on the same circuit in a regular receptacle (but not in the usual permanent wiring) then the problem is with that wiring - i.e., box it connects to, box in attic, etc.