I tried doing a search, but didn't find an answer, so here I am.

The random event's that come up (like the settle ten landmasses, build 8 swordsmen, what have you), is there a way to track what you have been given? or is it just you have to remember...

Quest-type ones can bee seen in the log -- Crtl-TAB or the little notebook icon in the upper left corner.

Is there an easier way to check how far you are along on a building quest other than checking city by city? (You can check on the unit quests by looking at the military advisor.)

Go to graph icon in the right conner of the screen. Open up statistics submenu and sort through however you like. You can try to mouseover in ctrl tab bar , I am not completely sure in works.