Google 'resettable thermal cutouts'. Do you also have a suitable battery management system to prevent over charging and discharging of the battery? That is important, as lithium batteries can catch fire or explode if not treated properly.
If you're planning to feed power back into the grid you would need the approval of your mains power supplier and your inverter would have to be type approved.
Sort of. For an NPN transistor, if the base is taken negative far enough (typically about 5-6V) with respect to the emitter then the base-emitter junction breaks down and current can flow. But the breakdown damages the transistor to some degree.
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Horses for courses? If the main criterion is pumping at maximum pressure and flow rate I would have thought a square-wave voltage excitation would be the most effective and certainly the simplest. If the main criterion is achieving non-pulsatory flow, then trapezoidal or...
For how much longer, I wonder? The trend these days, at least in commercial products, is to use surface-mount components rather than through-hole ones. The latter are getting more difficult to source. A tool which simplified the soldering of surface-mount components would be really useful.
Members may be struggling to work out what is connected where. You have the benefit of a board in front of you. It would help if you could highlight/annotate/label relevant points in an image. You can upload images to this site to make it easier to follow the thread.
If R1 and R2 are both made 100k that will certainly help, but the existing 100k should be reduced to ensure >2/3 of the supply voltage and < 1/3 of the supply voltage is reached when the button is pressed.