Re: inverter to AC breaker box (off-grid)
The reason you shouldn't connect the inverter to both sides of a 240 split phase panel is that certain wiring methods allow 2 circuits to share a neutral as long as they are on different phases, thereby allowing the neutral to essentially carry double the current that any give size wire would be rated at. By being on different phases, the neutral load would never exceed it's rating because when one phase was peaking at the top of it's phase, the other was at the bottom of the wave.
I am not explaining this very well, but I hope you get the idea.
Now in the real world, in a small off grid cabin it seems to me that this wouldn't be an issue assuming that you have wired your building and your panel to account for this rule. A small number of branch circuits carrying small loads shouldn't be a problem, assuming that the neutrals are wired properly.
I'm sure that some one will chime in with a better technical reason.
Please note, being a moderator does not add any weight to my opinions 300 watts Siemens/BP panels,plus a Sun 90,, making ~400. ~30 amps into Rogue MPT-3024, 450 ah of Trojan T-105, Morningstar ts300 inverter, a Tri-Metric meter.a collection of antique generators, plus 2 Honda eu-1000i's (also a BS2512 IX controller) and assorted other stuff!