Competition in EB is curtailed partly by money, yes, but more thorougly by the strength of the Eleutheroi, and the solid fact that, (Excepting on Very Hard difficulty) AI factions will always choose to attack the Eleutheroi in preference to a neutral faction.

As for trade rights, this is a very variable question, and it depends on your position and upon your settlements.

Click upon one of your settlements, examine it using the details tab, and see honestly how great a revenue that settlement generates through trade.
This will tell you whether to upgrade the ports in that settlement.
Markets should always be upgraded in any case.

Myself, I have conquered Byzantion as the Getai, and find it a brilliant trading city, in addition to Mytilene, Crete and Alexandria - the Crimean holdings of mine also generate a tremendous amount of mnai for their position.
However, inland, my holdings surrounding the Carpathian mountains and lining the Pindus Mountains are terrible in relation to trade, however they are all very capable mining sites, and so, through different upgrades, allow me more mnai than my trading cities.

It does very much depend on how carefully you manage your settlements as to whether the question you ask will bring you a great amount of money.

However, either in the beginning or at any time in the game, you may wonder, "Should I establish trade rights everywhere?", and the answer is always yes, because if you haven't got a trading city now - chances are you soon will, and you want the revenue it can bring then and there.

Trade revenue can be seen by looking at the financial tab also.

In my experiences, the trade in this game is now where near as important as in Medieval: Total War, in which your entire kingdom could depend on trade, and be ruined by a declaration of war.. I have never had a nation dependent on trade in Rome: Total War.