Thus Rome can dominate this era, what civ is the most dominant in each era?
Ancient Era, the Huns: Battering Ram, Horse Archer. Way too strong, it's ridiculous really.
Classical Era, Rome: Ballista, Legion are a dominant force as long as you mine the iron straight away. You're spot on about that.
Industrial Era, Sweden: Hakkapeliitta, Caroleans. Caroleans are just great. Hakka's not as much but when you pair their mobility with Caroleons then you have a very strong army for this time period.
I can't really think of others civs with units in the same era (are there even any others?). As for individual civs, just check out the individual units elimination thread and that gives you a pretty good idea.
However, I think there are a few exceptions:
Ranked 48/49: Maori Warrior. They are utilized to their full potential when you are playing a defensive, cultural game. Early on they destroy barbs camps like no other. When you unlock construction you build the moai and with the strength bonus you have a defensive fortress as long as your cities are built near the coast (with the moai culture/gold bonus, you
want to build near the coast). For the offensive game, they are not nearly as strong but it's great that their bonuses are retained to the end.
Ranked 33/49: African Forest Elephant. Slightly more understandable as they aren't as low but if I can recall correctly they didn't get much love. Feared elephant bonus, i'm a big fan of. Great generals bonus, i'm a big fan of. They do have one less movement than the Horseman but the significant strength advantage more than makes up for it. Just a very solid all-around unit for its time.