1) Not always. The engine kicks in when the gap between actual and desired temperature is great and the car needs to bridge that gap. As with heating, the impact on fuel economy is negligible, unlike when the petrol engine is driving the car!
2) I don’t see any reason why not but @kezo will advise. I never use the app. I just pop it on charge every time I get home, but I don’t have a cheap night tariff so charging any time makes no difference to cost.
3) In all honesty, charging away from home is utterly pointless (unless overnight at another residential property). Two reasons. Firstly you will pay more for the few miles that you’d gain than it would cost to drive those miles on petrol. Secondly, PHEVs charge at a very slow rate (mine takes about 6.5 hours to get 38 miles) so even a few hours will get you less than 10 miles EV range. It’s just not worth it and not cost effective either.
4) I’ve no idea but thankfully @kezo will.