Are there any splats that you would consider "essential" for fixing any issues in the 3.5 core rules? I've mostly forgot what problems 3.5 even has.
No, in fact, in general splats tended to make the problem worse. The core rules have essentially one basic problem. At some point, generally around 7th-9th level, spell-casters start outshining non-casters, and above 11th level, the game is entirely geared around spending caster resources. While a few splats attempted to 'fix' this by ramping up the power of non-casters, none really did a good job of it and most at the same time broke spell-casters even worse.
Also I'm wondering about the opposite side of things... Are there any splats (prestige classes, feats, spells, races) that I should be extremely wary of allowing?
In my opinion, all of them. In fact, the first thing I'd do is ban all non-core spells and all prestige classes. Play core classes only, and in general look at the 3.0 and 3.5 version of each core spell and make the weaker of the two the one that prevails in the campaign. If you add anything to core at all, and I'm not suggesting you do so, the only thing you should be adding is fixes for the fighter in the form of more powerful feats and options for the fighter and similar classes to try to stem the tide of spellcaster power a little.
Understand in core your main brokenness is Cleric, Druid, and Wizard and that the more spells you add from outside sources the more broken they become. In particular, the problem with these classes isn't just the sheer brokenness of them, but that with open access to spells each new spell in the campaign gives them another class feature and another tool to solve problems they previously couldn't solve.
However, if you stick to core, the casters won't be excessively broken unless they start abusing shape-changing spells and summoning spells. (Watch in particular for Polymorph Other abuse and Planar Ally abuse.) They'll just be slightly broken at that point. Also, if you stick to the 3.0 versions of many spells, even that problem is mitigated somewhat.
I'm not particularly concerned about things that are a little over-powered or strong, but those few rare things that in your experience completely broke down the game.
So, if you are wanting to keep your ban list small, ban anything and everything that lets you get free metamagic or greatly increases effective caster level or spell DC. Ban Robilar's Gambit and Assault Trooper. Ban Natural Spell (which moved into core with 3.5 because 3.5 had to break everything). Ban Ur-Priest and Frenzied Berserker. Ban Psions, unless you ban Wizards and Psions are replacing wizards. Ban everything from BoED, BoVD. Ban Tome of Battle
unless you ban all non-Tome of Battle non-spellcaster classes and just run a Tome of Battle centered campaign...
You know, seriously forget about it, your game will just be better core only. If it isn't going to be core only, then the best policy would be to ban part of core and use some replacement classes to achieve balance. However, IMO that runs into two problems. First, you tend to end up in a place where nothing is broken because everything is. And secondly, you tend to end up with parties well optimized for going supernova on opponents and most combats are effectively over by the middle of the first round, which was kinda what you were trying to avoid by putting bans into effect in the first place.