The only thing I think needs some further adjustment is merc stats. Lower merc cav hack armor a bit, and merc infantry pierce armor a bit. Beyond that, they're fine.
What do they contribute to the game? They make playing or playing against Egypt very different from the other civs. Many of the same strategies won't work, or won't work as well. Sending a bunch of rams in the back way to "sneak attack" a TC is much harder vs Egypt than other civs. Succeeding in a direct attack on an Egyptian TC is hard too - it's not really something you'll want to do a lot. The bottom line is - you can't really beat an Egyptian by taking his TCs away. You take his TCs away when you've beaten him. Two different concepts, which actually essentially makes for a different strategic profile from the other civs. Against Egypt, controlling their gold supply is vital - more so than for any other civ - if you want to beat them. Otherwise, you need to lure their army out somehow and beat it where it can't get significant merc reinforcements, and keep forcing exchanges like that. Egypt has a home ground advantage that the other cultures generally don't - it's a design thing, and it's one of the things that really sets them apart strategically, not just in terms of having a different look.
And if you do assault an Egyptian TC anyway, you probably need to use most of your army, along with a good deal of siege, to pull it off.
I think when people look at mercs as a problem, they jump one step ahead in the equation - it's not mercs that are the problem, it is the other balance issues that enable an Egyptian to get to the point where they can have a full army + mercs that is the problem. Most specifically, Set's animals + god powers, but also Isis's god powers and hardcore defensive profile. For Ra, which I play quite a bit, mercs are currently much harder to get out - in order to be able to survive in the first place, you need an army, which locks you down, and forces you into equal exchanges - unless my enemy sits back for a while without engaging me somehow, I very rarely have the budget for more than 1-2 mercs - as an emergency thing. That's how it should be. Being faced with a lot of mercs against an Egyptian is either the result of other imbalances, or sloppy play on your own part, not forcing the Egyptian enough. That said, I still think merc stats should be reduced a bit.
The cost of mercs is quite high in general as well - 120 or 90 gold may not sound like a lot, but once 2-3 are made, you should generally be able to get an upgrade ahead, whether economic or military, or be able to plop down one of those equally pop-free hill forts. For this reason, in fact, forcing or tricking an Egyptian into making a lot of mercs can be a good strategy sometimes as well - at least if the game hasn't gone too far.
Now, making mercs cost favor is one of the most horrible ideas I've heard. There's not a shred of "mythologicalness" about them. What would suffer from making them cost favor isn't the Egyptian - he'd just make mercs, and not make any MUs at all - same end result, same number of mercs, fewer of the really fun units. Fun drains right out of the game - for both sides. If you want to increase their cost, just bump the gold cost a little instead - merc cavs seem a bit underpriced when compared to merc infantry. You'd be seeing significantly fewer merc cavs at 150 gold - not just 1/5 less, either - their general level of cost effectiveness would decrease as well. At 120 gold, their cost is similar to that of a jarl or hippikon, though with somewhat superior stats.
Then you might say "but what about the late game?" Well, late game, Egypt gets spanked by Greeks, and Norse aren't really far behind either. Egyptians generally have 1 - one - unit which can stop hoplites or ulfsarks - and that unit gets shredded horribly by building fire, toxotes fire, or ballista fire - not to mention that the other civs are capable of fielding much larger quantities of MUs, some of which are very hard for Egyptians to stop (like colossi, fire/frost giants, even chimeras). Being able to get ballista towers and mercs is pretty much a must-have equalizer at that point, since Egyptians would get owned hardcore otherwise. Egyptian troops in general aren't as cost- or pop-effective as those of the other civs.
One other potential problem I could see is - once an Egyptian captures a TC, he holds it. I think this is perfectly possible to overcome, though - if he takes one more TC than you, you're effectively up 400 gold or more (while the TC cost may look more expensive for the other civs, the real difference, if any, is smaller than it looks - wood is cheaper than gold, and Egyptians have long build times, and using the pharaoh to reduce that costs resources elsewhere) - if he's able to afford mercs beyond that, I think you're doing something wrong somewhere. Against Set, of course, or even Isis to a lesser extent, this can still be a serious problem - but that's due to the god power/civ imbalances, much less than it is due to the mercs themselves.
Removing mercs would be a hasty, ill-conceived move as well - it'd be the removal of something unique from the game, that potentially makes it fun to play. You might as well ask to have Norse infantry building removed and have them build with vills like the other civs. Actually, that's a very good parallel, as currently, Norse can spam shooting buildings like nobody's business - which is at least as bad as mercs. You know, pop-free map control. Again, a sort of necessary evil, though hill forts and Norse towers should be made somewhat more expensive in general - +50 wood for Norse towers and +100 wood for the hill forts, and they'd still be the cheapest towers and forts, though they'd be closer to Egyptians or Greeks in cost.
You can't really discuss mercs without pulling in the whole picture, and in the whole picture of things, I think Egypt is somewhat strong, but when you look at Ra, you see Egypt in its "purest" form - without must-have god power crutches and animals - and Ra is currently one of the weakest civs across the board.
If you want to address this problem, you should start with 1) Ancestors and 2) Set animals. Once those two are balanced in general, you won't see nearly as many mercs - it just won't be affordable. The occasions when the Egyptian makes them anyway, it'll be much more likely to be a desperate measure than something to own you with.