You don't get "commerce" by trading (it's not like the caravan thing in civ1,2). Commerce is always produced by citizens working on a tile. Then the adjustment of tax/lux slider does a partition of your total commerce into gold/science beakers/entertainment (look at the city display: there you can see how commerce is devided according to your settings).
By trading with other civs, you can only get gold, so to speak -one part of commerce-. Of course, if you get a nice gold-per-turn deal, you might increase your science rate, i.e. spending more commerce on beakers while your opponent pays gold for that *missing* tax-used commerce.
Referring to "what you have read" (I don't know for sure

), you
can set up a deal that gives you gold, e.g. trade away a lux resource. So trades don't give you only resources. BTW, the happiness thing applies when your opponent gives a lux to you.
You don't need an embassy to set up trade. You need at least a road connection (e.g. for continental trading) to their capital.
Overseas trading need harbors or airports(and contingently connection to capitol via road). Furthermore, overseas trading need certain techs: map mapping allows trade routes over coastal tiles only, astronomy (or owning Great Lighthouse) for sea tiles, navigation/magnetism for ocean tiles. Either one trading partner must have these abilities to set up a deal.
note: trade routes can go via 3. party roads, harbors, airports, culturally covered coast/sea/ocean tiles if no trading partner is at war with that 3. party. Such routes can cause a rep hit when war breaks out and they get shut down.