A bit more info on the vias to join the sides thermally:
https://www.edn.com/electronics-blogs/t ... ermal-vias
2. I'd put a pull down on the mosfet as a matter of habit. Thinking about it more since my earlier emails to Chris, the failure mode I'd be concerned about is the micro dying and leaving the mosfet gate undriven - it could then float up to the point where the mosfet turns on and flattens the battery. If someone is away or doesn't monitor their battery regularly (eg for a house) they might not notice things have gone wrong until it's too late and the cell is dead. The micro could fail in other ways too (pulling it high or low) so it isn't a guarantee, just risk mitigation.
However... given you have the holes there for a through hole mosfet it would be possible to fit it later on using a through hole resistor.
3. The cap is just there to stabilise the power supply, the specs shouldn't be too critical. Colleagues who know more than me shy away from Tantalum caps because they can fail "short circuit" (bang) - electrolytic caps tend to fail open circuit. Ceramic caps have improved a lot over the years - why not go for a 10uF 10V 0805 ceramic cap like Seeed 302010158 ? You could put a through hole set of pads (2.5mm pitch) next to it for those who can't easily source the SMD part and want to use a through hole electro. Plus because it's so much smaller you could put the mosfet pull down resistor in

This looks like an interesting source of info for the differences:
https://electronics.stackexchange.com/q ... ces-in-use