I'm getting the impression that Agility is key to winning the battle while Range allows you to get more into the battle. I guess airfield placement/size is also a factor for getting planes into the fight.
I think you've got the idea. Agility counts, but so do numbers. So the decision at that point isn't necessarily a black-and-white cookie cutter. If you're just fighting in Western Europe, you're not going to hit the same range problems you get in Russia, Africa, the Pacific, or China. So you might prefer agility if you're Germany, range if you're the US. Agility makes your 1v1 fights better; range makes you more likely to have some 2v1s.
I'm still hazy on how to get the armor bonus and clueless on how the piercing vs. armor dance goes.
Check the definition of the stats in the
Land Units section of the wiki (and hope that text is still accurate

.
Armor is essentially a binary comparison versus your target's piercing. Their division can either yours theirs, or it can't. If you have armor higher than the enemy piercing, good things happen. Most importantly, you halve the number of hard and soft attacks against you. Offensively, higher armor than enemy piercing also helps because it increases your number of attacks, and also makes each attack do more org damage. (That's what the d4 increased to d6 business in the wiki is about.) Piercing is the opposing stat. If you get higher piercing than your target's armor, they don't get those bonuses. Battalions with high Piercing also tend to come with a large number of hard attacks, making them good at killing enemy armored vehicles.
A really important point is that piercing and armor are not static values. It's easy to look at the stats of one anti-tank gun or one tank, and say, "hey, that can piece me", or "nope, I'm safe". But the armor and piercing values we're talking about are the ones the entire division has, at the moment each combat round is fought. So, the tank's armor value is getting diluted by the infantry or support companies in the division (which have their own value, especially Org; max armor isn't the ideal). Similarly with Piercing. That's calculated as 40% of the highest Piercing battalion, plus 60% of the average of the rest. So just having one AT gun or TD in the division helps out of proportion to the total numbers, but it's still diluted somewhat. (In theory, the field commanders put those assets where they can do the most good, not randomly versus enemies. But there's only so many AT guns in the division to share around.) And both those values depend on current equipment totals. If you lose tanks or AT guns in combat, the armor and piercing values change. So in the field, combat losses, supply, and attrition are going to affect the actual values of your divisions. Carefully designing a tank division on paper to have 0.1 more armor than the enemy's best AT gun doesn't guarantee it can't get pierced in any particular fight. Make sure to keep an eye on the combat readiness of those units and their opposition (as best you know it).
Ideally, you don't need any more Armor than your enemy has Piercing -- but knowing exactly what that number is in the face of changing templates and losses isn't an exact science. Also, since HoI4 is a grand strategy game and not a tactical battle game, it's something you need to consider from a long-term perspective. It takes time to do research and get equipment produced and deployed. So, what's your current situation? Nice edge in armor? Maybe you're good for now -- but what do you expect the enemy to have deployed in a year? How long is your edge going to last, and when do you need to start research to make sure you have new equipment to retain that edge? Enemy just started deploying some new AT technology, or you've switched to a new war against a new enemy, and your LARM can't kick ass the way it used to? Obviously you're going to want to get back in the armor game -- unless you're planning on reducing your offense and digging in, or your strategy is built more around airpower, in which case you might do just enough to keep up rather than launch a massive effort to regain armor supremacy at the cost of other branches.
So yes, "it depends". That's a good thing from the point of view of having an interesting game. For any given patch, some things will be OP, or broken, and shift decisions accordingly. But the game isn't really as simple as "just stamp out nothing but this exact template maximizing this particular state for the entire war". Even if it works, it's boring. So try different approaches, and see what you can to work and what doesn't. Understanding how the stats and systems interact will give you a better idea of how you can shape those different approaches -- because it's also not true from the gamist balance perspective that any equal-cost approach should be and is equally valid and equally likely to win all your wars.