On higher difficulty levels (immortal/deity), is there ever a good reason to mass attack with, say, infantry and artillery? or fighter planes? or non-nuclear missiles?
In later eras it is very important to fight very quick wars because of AIs having better production and also better connections.
Therefore, Cavalry + Airships.
Later, to kill infantries transition to Cavalry + Bombers is very good as Cavalry is still useful.
Infantry + Arty can be good for choke points and when you need to deal with SoD, but for taking rest of the territory, I would advise fast units.
For dealing with lot of Coastal cities, Fighters + Marines works well. You get high mobility from Carriers (as they are the slowest). You can produce Infantry before getting Industrialism. Infantry will clean-up rather well. For inland cities you are not forbidden to carry a bit of Cavalry or Tanks (you do have Industrialism anyway) as one movers are too slow.
Tanks + Airships (as for Bombers you still require Flight and Radio, both very expensive), and then replace Airships with Bombers. This tears through anything.
This is where I stop normally.
Getting to Nukes is easier than getting to Modern Armor and Mech Infy. If you must resort to this, game wasn't played all that well, but the win still counts.
For reaching irradiated inland cities, Paratroopers work well.
Sometimes, I'll have ton of cities and tons of research and get to Modern Armor and Jet fighters, but I'll never wait for them to start a war.
Units I don't find useful at all for Ancient start deity games: Knights, SAM Infantry, SAM Artillery, Ship of the Line, Mobile Artillery, Stealth Bomber, Jet Fighter, Attack Submarine, Stealth Destroyer, Anti-tank.
Or is the post-cannons-cavalry-riflemen-era the point where you just beeline nukes or consolidate your diplomatic situation and go for space victory or diplomatic victory?
For safest conquest/domination victory, should you just stop teching entirely once you hit rifling and steel, and take over the world with mass drafted rifles, spending gold on upgrades or unit purchases through universal suffrage, and leaving tech at 0%?
Depends, is it achievable? If it is, then yes. But Rifles and Cannons are slow and slow to come to the front line. Then some guys will get Railroad and will flank your Cannons with Cavalry and also kill a lot of your stuff as they will also have 4 Airships per city, like AIs do. I believe you'll still need to get Artillery in such games to stay competitive. Vassals can help you get there even if you cannot maintain a research slider.
At what point do you go for nukes and at what point--if any--do you go for later military units other than nukes?