Yes, false life and Heavy Armor Master work together.
Temporary HPs function exactly as normal HPs, except that they are designed to be lost first, before applying residual damage (if any) to your normal hit points. They are not to be mistaken for damage resistance, which is the ability to halve the damage taken in some situations, or damage reduction, which is the ability to reduce the damage taken by a fixed amount. All of them, however, should stack.
Player's Handbook (p.198)
When you have temporary hit points and take damage, the temporary hit points are lost first, and any leftover damage carries over to your normal hit points. For example, if you have 5 temporary hit points and take 7 damage, you lose the temporary hit points and then
take 2 damage.
If you possess some form of damage reduction, like the Heavy Armor Master feat, or some form of damage resistance, like the rage ability, you apply these effects first, just like you would when receiving damage normally (remember that, as per the rules found on p.197 of the PHB, damage resistance is the last modifier to be applied to damage taken). In effect, temporary HPs are a way to magically augment your ability to take punishment.
Contrast this situation with an abjurer's Arcane Ward. In this ability's description, the ward is described as a separate construct having it's own set of HPs, rather than giving you temporary HPs. As such, one could possess both an Arcane Ward AND temporary HPs (although temporary HPs from different sources still wouldn't stack), but the ward shouldn't, at least according to RAW, benefit from damage reduction or resistance effects. Indeed, it is not you taking the damage, at least until the ward breaks.
No feature which grants new spells counts against your spells known, unless it explicitly says so. Spells Known for the Warlock is exclusively for the Pact Magic mechanic, which this is not a part of.
Sure, it can't hurt.
Assuming that the player agrees (and I see no reason for them not to), yes, you can just assume that a warlock with that invocation starts with their buffs on.
I played a 3.5 game where all of us were using highly optimized, self-buffing characters who would stack buffs on ourselves before every combat. It was a waste of time for everyone to announce their buffs at every encounter, so the DM just assumed that we had them on at the start of each non-surprise encounter, and we subtracted the resources ourselves. We never had any issues with this system.
Just don't forget about it--having a spell constantly on like that means that they will show up on detect magic, for example, and there might be situations where constant obvious spellcasting might draw unwanted attention.