I've never had a decked-out hero. I can get a few decent items every once in a while, but tend to lose them fast. So, I tend to stick with mediocre armor even at hero, unwillingly.

I feel like there are four main things to cover: hp/mana, stats (str, dex, etc.), saves, and damage roll. It seems the best I can do is have two out of four of these categories covered with easily accessible gear. For melee classes, I usually do stats & damage or stats & saves. For casters, I usually do hp/mana and saves. For casters, that seems fine, but I'd like to cover 3 or 4 of the categories with melee classes.

I consider covering a category to be a couple hundred for hp/mana, 8+ for dex/str, 25+ for damage, -15 to -30 for svs.
Are these numbers easily done for 3 or 4 of the main things I named?
What sort of numbers do other people shoot for with gear at hero range?

Could the item search on the webpage be expanded to include some more limited gear to help finding things? Would that be a lot of work?

If you just want to get an idea of what's out there, look up PBFs for whatever build you're playing and check the "Best Set of Equipment" section for the ones that have it.

Keep in mind, a lot of the idea of gear is that there's generally NOT
a perfect set of armor you should go get - or if there is gear that's
obviously the "go-to", it's very limited and fought over. I tend to
feel it makes CF more interesting, as you don't have every character
wearing the same gear, and you often have to make choices. Do you
keep item A, or item B? Or maybe you carry around an extra item to
switch out? Is it something you can get easily when you need? Etc.

As far as simply mechanically, I expect limited/not limited is a
pretty simple way to make the item search work. "Some" limited is
probably far more difficult to do.

Personally, I'm generally shooting to cover my own needs as a "general"
set (stats, dam as a warrior - hp for a mage), and then consider my
main difficult or frequent enemies secondly as to what gear I either
commonly wear, or keep as a "swap out" set.

The item search doesn't have those items specifically to encourage more exploration on your part. They're not on there by design.

All your questions really depend on what you are playing/your playstyle.

Yep, people gear differently depending on how they play, there is not really a best set, it varies by preference. Of course there are some very great gear pieces that any char would want, but for the most part you're amassing noteworthy and occassional famous/legendary pieces.

Most mage players only care about hp/saves, but for others some str and dex can be useful to cover stat loss that may impair movement when they need to escape and they are willing to sacrifice a little hp/save to cover this.

I geared all my bards in the same way since they were w-elf. 300/400 hp, 50 svs, +10-15 dex. Nothing else really mattered, since I tended towards very light weapons that would not be str disarmed and I had songs to provide extra stats across the board.
I could manage this kind of a set without having anything that was overly high end, and I could obtain it all solo.

I always found knowing where to find middle of the road gear that was usually not over limit and being able to acquire it solo to be one of the most useful things in CF. It just takes time to get this knowledge, and if you rely primarily on PK for gear you are at a severe disadvantage. PK is how you turn a good set to a great set, but being able to build a good set solo is a good foundation and enables you to better build a stronger set over time through PK (and higher end exploration). Also, it makes being looted less of a set back when you die.

Encourage people to show you gear locations IC and learn how to acquire it is the best advice I can give. Better to know where gear is from and get it yourself then to be given a piece of gear that you'll never find again if you lose it.