I did the same thing in my set of house rules, with the exeption that good and medium BABs were only worth a +2/+1 boost.
I think I like the better definition between the BABs, but I can see why one would like the other way too.
#1 can probably be solved using one of the many alternative systems floating around here (e.g. multiple attacks incur a -2 penalty on all attacks).
That seems like a good pay off and then say that, every 6 levels you can do it? 6th/12th/18th --> -2 for 2 attacks/-4 for 3 attacks/-6 for 4 attacks?
#2 wouldn't be a problem if the level 1 boosts weren't as high.
Perhaps the average of all of your classes, so a fighter wizard would have +2. A rogue/fighter would be +3 then and a fighter/wizard/rogue would be +2 and a fighter/cleric/rogue would be +2 too. This seems fair.
Saves are non-stacking bonus for the classes So if you are a cleric/rogue you get +2 to all three.
#3 isn't clear to me, can someone explain?
These are Pathfinder defense and offense, basically "grapple" expanded to include bulls rush and other things.
#4 is a nonissue; you'll have to cut back some monsters' natural armor (which often is ridiculously high anyway).
Or leave it as is
What exact problems are you trying to fix with this suggestion?
The main thing is the separation of BABs and saves at mid and high level. A fighter has +10 more to hit over a wizard and a poor save is +6 more over a good save. Secondarily their is a multi-class funkiness where you can get super saves or horrible saves and BABs. The BAB seperation is not that much an issue however the saves is. Spell DCs way outstrip, the power level of saves at the upper levels (and even low mid at times), when you compare those saves to "good saves" you get a decent chance of success, when you compare to a poor save they have almost no chance of success.
Example:
Wizard's 9th level spell
16INT, +5 for level increase, +5 Book, +6 headband, for a 32 (+11)
DC 30
Fighter's Will save
+6 Base save, 14WIS (+2), +5 cloak, for a total of +13
They need a 17+ to save
With this 1/2 level thing they would need a 13+ still difficult but considering the maximized nature of the Wizard.
I don't think attack bonus really is much of a problem. If your character focuses on doing damage with weapons, you make sure not to dilute your BAB. If you don't focus on doing damage with weapons, then by the time you hit level 10 or so, it doesn't matter that you aren't very effective with your crossbow--you have spells for that. End result, characters who need to hit AC can hit AC. Characters who can't hit AC don't care. Of course, since 3.x and pathfinder are not 4th edition, monster ACs don't necessarily scale with level so your poor BAB wizard probably can hit the ogre barbarian with his crossbow. If he doesn't, it's because he's got two empowered scorching rays, a quickened ray of enfeeblement, five magic missiles and a wand of enervation. Why would he bust out his crossbow again?
Agreed, BAB is not the main interest with a rule like this. However, Multi-classing can cause some wonkyness in you BAB score and this would fix that. However it screws with feats, and PrCs. It may be simpler to just adjust saves.
Saves on the other hand are a bit of a problem--mostly because of multiclassing which tends to yield either unbalanced saves (the fighter/barbarian with a Fort that can't fail and Ref and Will that can't succeed) or massive saves (a monk/fighter/paladin/pious templar with +17 as her lowest save at level 10, evasion and mettle who says, "is there a save? OK, I'm effectively immune.") Divorcing saving throws from the class mechanic and pegging them at 1/2 level fixes this problem and renders it much easier to create reliably fair DCs. Just give a +2 nonstacking "good save" bonus and multiclassing works a lot better.
You have a good understanding of the problem!