What's with clicking on workers when they are 1 turn away from finishing what they are working on, then clicking again, completing the build? Is this something that actually makes a difference? Don't workers finish work before calculating what a city gets?
Yes, just a micromanagement habit, so i can see the "clear" picture before my turn ends. If you chop the forest that way you will see the exact number of hammers you gained and be able to switch the current build, maybe, or adjust by selecting a different tile to end a turn with. This isn't game breaking or decisive, just my way of immersion, i guess.
And, of course, every little bit counts.
Is the idea that hammers never carry over, so you want builds to be exact if possible
The idea is that hammers do carry over, so, most of the time you will enjoy the overflow. Food carry over as well. Hammers also degrade after certain duration of inactivity. I think it is about 20 turns until you start losing hammers.
If Rome doesn't have Notre Dame + Chicken Itza, what do you do about happiness during the rest of the war?
That's the trick, always. Plan your conquest path through unique luxuries, happiness wonders and gold - rich cities. If you can't, you can either risk doing it in unhappiness, or, go for science vic., or, wait until you have enough happiness to proceed.
I might have skipped past this a bit -- did you have a spy in Rome to see what they were up to? How did it help?
Sorry, the whole thing was a while ago, i don't remember.
I probably payed Augustus to attack few civs, that usually reduces their army. Especially, if they fail.
Incidentally, i was recording it as well, so, now you can have a look how domination game on Saturday morning may look like. 
