Mines: they are not represented as anything, it happens every turn (and you get an event message from it), it is just what the ship landing on the red hex represents, but the event is the damage to the ports and such.
1. HQ: not really, but. The HQ needs to stay as close to the supply as it can, esp in enemy territory, where you won't get 10 sources, so you capture a town, it gives you 5 supply, HQ sets up shop on it and gives 8 towards the units, great, you move one hex to a mountain, one distance, and +2 movement penalty, so a total of three lost supply, the HQ boost 2 to 5, same as the town, the HQ does nothing. HQ should be either in town, or on a road, to help it getting good supply, and good way to distribute its supply boost. Of course special situations can happen.
2. Reinforcement: damaged unit lose power, 10% per lost str, also tends to go for lower morale and readiness, further reducing its effectiveness, so in most case, a 10 unit is better than an 8 with one star, and you really cannot keep your xp. in most case, you units will have little to no xp, as the war rages on, maybe your art, or protected bombers can get some xp, but you will take damage, and you want it to be reinforced.
3. The AI has some strict scripts, but also very flexible, if you are defeated in the med by the UK, they will land before the US arrives, if you defeat them in the med, than the UK might skip Italy and focus on Normandy, and if the US AI agrees, it will do it earlier than happened, or they will do multiple small landings to harass you. If the SU struggles, they try faster landings, if SU doing well, they go for japan a bit more.
4. Europe Fortress: consider most hex unprotected unless it has a corps on it, garrisons, coastal guns and such just a speed bump for a well done landing.