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.

Chain Shirt

A chain shirt is the right choice for almost every character, sadly enough. It provides a fairly big wodge of AC, minimal armor check penalty, and has a nice, big maximum Dexterity limit.

In theory, a lot of characters could swap this for mithral full-plate if they have medium armor proficiency, but in practice it’s usually not worth it; armored AC just isn’t very valuable, and that’s an expensive armor. But the point is moot anyway, since rangers lose class features if they wear armor heavier than light.

Anyway, in general, the only reasons to use another armor are if you

1. have extremely high Dexterity, or
2. have quite-low Dexterity and heavy-armor proficiency and are not an arcanist, or
3. have class-specific requirements to use something else.

For those with very-high Dexterity (expect to reach 24 or higher), a chain shirt is limiting. In that case,

• Spidersilk, +3 AC, +8 max Dex (Underdark)
• Nightscale, +2 AC, +10 max Dex (Underdark)
• Gnome twist cloth, +1 AC, no max Dex (Races of Stone)

For those with quite-low Dexterity (maybe 15, but probably not even then; mostly we mean 13 and less) and heavy armor proficiency, full-plate may be better. You get +4 armored AC relative to the chain shirt, and since your Dexterity isn’t that high anyway, you don’t lose AC. But armored AC is not particularly valuable, and that −5 (after masterwork) armor check penalty is prohibitive for a lot of people. Like the high-Dexterity folks, there are additional armors in supplements, such as Races of Stone’s mountain plate, but the armor check penalty problem becomes even more severe, plus many of these require Exotic Armor Proficiency, which is just not worth it.

Almost no one should use any armor between a chain shirt and full-plate. Breastplates have double the cost, double the penalty, reduced max Dex, and heavier weight, all for a measly +1 armored AC; not worth it at all.

Obviously, if you have a class feature that says you must wear some other armor, or you can’t wear certain armor, you have to pay attention to those and they can change any of the above. The monk is the obvious example. But as noted above, in your case, the ranger itself does have class-based armor requirements – specifically, to wear light armor. A number of their features don’t function in medium or heavy armor.

If you have high, but not very high Dexterity, in the 20-23 range, you just have your chain shirt made out of mithral. It’s relatively cheap, and has very few downsides. Mithral can also dramatically improve full-plate as an option, improving that −5 to a far more manageable −2 and allowing your Dexterity to get up to 17 without the armor limiting you. Whether this is actually worthwhile is debatable; armored AC really isn’t worth much, and that’s a lot of gold for +4 armored AC over a simple chain shirt.

Also, avoid wild

The wild armor property is very expensive, as a +3-equivalent, and it only keeps the armor bonus of your armor, none of its special properties. The special properties are about 85% of the point of having armor in the first place; this is not a good choice. (If you do go with wild though, you might as well get the biggest, heaviest, highest-AC armor you can find, as recommended by Scrollreader.)

Wildling clasps, on the other hand, limit you to forms that could reasonably wear your armor. As a master of many forms, that’s a pretty dire limitation. No good.

But the beastskin special armor ability allows you to continue to wear your actual armor, complete with its other special armor properties, on any form you like. It’s priced as a +2-equivalent and requires an extra use of Wild Shape every time you want to use it, which is brutal, but by keeping the wildling clasp you can limit how often you need to do that to only those forms that cannot wear your regular armor, conserving Wild Shape uses. This seems to be the best solution.