It seems like a lot of people posting here either haven't played for long or don't remember a few patches ago. Iron units used to be almost worthless. Horsemen had strength 12 and horse units didn't get a city attack penalty, meaning with move-attack-move you could pretty much win with nothing but horses. If someone got stuck with Iron iron and no horse, they would think "well at least I can build some Trebuchets and defend my cities, but I'll never be able to conquest because I can't build knights."
All that happened was that the dial got turned too far back. I think the balance would be pretty much fine if the city attack penalty were lessened, Knights and Cavalry were brought up to 4 movement, and we lost the atrocious "if a Mech Infantry used to be a Spearman 3000 years ago it still has +100% vs. mounted" mechanic (with the exception of unique promotions from UUs).
Counter units aren't that big of a deal because of how % modifiers work. Yeah, a level 1 pike will do well against a level 1 knight with no other modifiers (20 strength vs. 16 strength), but once you add promotions, generals, and more modifiers the matchup swings in favor of the knight. Say each side has:
Shock 3 and in open terrain (+65%)
A Great General nearby (+25%, or is it 20% now)
1 flanking bonus (+15%, I think, or maybe 10%)
1 discipline bonus (+15%)
Heroic Epic (+15%)
I may be off with some of those; I haven't played much recently. Either way, it works since each side has the same % bonuses. I'm going to ignore open terrain defense because that could go whichever way, but it's more likely to come down on the side of the more mobile unit, so take that into account.
Anyways, here's how their strengths end up. They're additive, not multiplicative, so they don't modify each other. They just all add a % of the base strength:
Pikeman:
10 Base
+10 (vs. mounted) = 20
+6.5 (Shock 3) = 26.5
+2.5 (Great General) = 29
+1.5 (flanking bonus) = 30.5
+1.5 (discipline) = 32
+1.5 (Heroic Epic) = 33.5
Knight:
16 Base
+10.4 (Shock 3) = 26.4
+4 (Great General) = 30.4
+2.4 (flanking bonus) = 32.8
+2.4 (discipline) = 34.2
+2.4 (Heroic Epic) = 36.6
Yeah, that's some counter unit. That's not even taking into account the knight's move-attack-move or greater mobility.
Anyways, this just goes to show the problem isn't counter units. I'd also say that adding a bunch of unit categories to complicate the counter system isn't necessary. This is because, IMO, the problems are as follows:
1) Horses were over-nerfed in general
2) Because of the scale, Knights and Cavalry are too slow to make up for their weaknesses
3) Strategic Resources are too easy to come by
Item 1 I've already explained, but 2 and 3 will take some elaboration. I understand why mounted units have the speeds they have; that doesn't mean I agree with it. The reasoning, I'm sure, is that light cavalry (horsemen/lancers) should be faster than heavy cavalry (knights). Cavalry should also be slower because the point of the Lancer is to be able to counter cavalry and the occasional odd knight. Plus, having horsemen/lancers be faster than tanks wouldn't make sense (excluding Companion Cavalry; who knows what those crazy Greeks are feeding their horses?) So the scale of the game dictates that knights and cavalry be movement 3. Which kills their utility, which is the one thing they have going for them. Also, if you retool unit speed, cavalry soft counter ranged units better than they do now, so you have balance there without having to add hard counters. You could even buff ranged non-siege units a bit.
As for point 3, the scale of strategic resources further destabilizes the game. It's not necessarily that there are too many on any given map (although IMO 6-yield iron should be reduced to 4 or even eliminated; call me crazy), but rather that if you want 20 iron units, you can get 20 iron units. If you want 20 knights, you can get your 20 knights. Either through trades, settling, or city-states, but most likely through all three. That pushes all other units out to the margins in all but a few extreme examples. Plus, it kills realism. Warfare would have looked quite a bit different if every man in every army in history could have had full plate mail on his back, 5 feet of steel in his hands, and a horse.
Now how does this tie into my main point? In two ways. First, the game can't both be balanced and realistic like that. Either the units are mediocre and it's unrealistic, or the game is realistic but strategic resources dominate – assuming the longswordsmen are supposed to represent the later renaissance when they figured out how to angle plate mail to deflect blows. Second, the whole game has been balanced (albeit perhaps not that well) around this abundance of iron and horse; the less-than-stellar state of mounted units right now is a direct result of them running amok earlier and being brought down. If they could balance the game around expecting a renaissance army to have a combination of 6-10 horse/iron units, they could make those units stand out but still need support. That is, if combat against the AI weren't so easy, but that's an entirely different subject.
On an entirely different note, muskets are fine as they are. Well, at least in combat strength. People often seem to get this impression that someone fired the first portable firearm and everything changed. That's not true at all. Early firearms were garbage. They often complimented other armaments (as with tercios, dragoons, etc.), but they sucked. The reason they caught on was because you didn't have to spend your life (and fortune) figuring out how to use them. You could take a bunch of townsmen, hand them muskets, and tell them to pull the trigger. You now have an army.
Civ V tries to emulate that by giving them the same strength as longswords (less, until recently, and IMO it should stay less) but without a strategic resource and at a hammer discount. And that's fine. You can argue the fine points, but the musketmen are obviously supposed to represent the weapon/formation's early days, and it does a fine job of that.
Now for a really misrepresented unit, try the Pikeman. Pikes were fearsome in their time, and not just against horses. A pike formation could hold its own against just about any other melee formation for centuries. The problem was that they were slow, they were cumbersome, and they couldn't adapt to advancements in ranged weapons, against which they were completely exposed. So if you really want to find a unit that is poor in gameplay (especially with the problems I pointed out in not countering units) and lacks realism, take that one up. You could easily make it a strength 12-14 unit with a 100% bonus against mounted and a 50% defense penalty against bombardment and call it a day. If you wanted to get really ambitious, you could even weaken them per unit flanking them. You'd have to rebalance the other period units around them, but that's no problem.
But muskets? Muskets are fine.