Heavy armor and light armor (the very first perk you take with 5 ranks each, I believe). It just causes your armor rating to be boosted with each rank. I was considering maxing both out.

Depends on your build and if you want to use some pieces of Heavy and Light Armor, especially if you like the looks of it.

Each rank will boost the rating. Most of the other perks in both trees will not work if you mix and match.

You might want to use the mages armor spell to help with boosting the armor if you need it.

Smithing can also boost your armor quite high.

It depends on the character. I try to put perks where I think my character would put perks, if they knew about them.

Most of my characters stick to one weight of armor or the other. In these cases I naturally choose perks that apply to the armor they are wearing. But if a character wears both Light and Heavy Armor then I will choose perks in both. And, usually, I will eventually max them both out.

I only max the first perk on one armour skill, not both

if you do crafting, just one rank in the first perk is enough to get to the armor rating cap of 667 (80% protection, 567 displayed AR, 542 displayed AR if using shield)

There is no armor cap except when it rolls over. There is diminishing returns with armor rating.

But I have noticed when playing on both the PC and 360 that I take less damage with gear that is way over 667 then with gear that has only 667.

Yes, several people have noticed (and tested) the same thing, and we've discussed it elsewhere on this forum. It appears that the rate of increase in damage reduction drops off fairly rapidly above the so-called cap, but it continues to increase.

Maybe at one time there was an armor cap and maybe Bethesda decided to do the diminishing return. Maybe when they decided to add Legendary difficulty. I never did that much smithing until Legendary came out