Could you maybe write a guide as to how you set up and abuse deity AI to piggyback them for cities etc? AFAIK that playstyle is fairly unique to you and I've seen you mention it lots over the years. Things like tech priority are useful, but the real interesting points would be what kinds of units you use and how/when you know to vulture cities. You were one of the earlier advocates of deity medieval that I recall and I feel like that's still a missing part of my game outside of the engineering bulb rush.
Sorry for the long delay, I had somehow missed this.
First the boring things: Part of the effect is economic. On Monarch and below, the AI can't keep up if we're teching hard. Fewer trade opportunities, fewer discounts, fewer applications for espionage (if we cba).
This starts with something as basic as the Alphabet - I like to trade for it, but on mid levels the AI can take annoyingly long to get it.
If we want to go light on wonders (generally agreed to be faster), high levels allow us to use them for failure gold without having to build them ourselves. This can be considerably more efficient than gold via commerce.
If we go heavy on wonders, we're likely to beat the AI narrowly a few times, and full AI coffers make for good trade opportunities.
Now to the bits you were interested in. Yes, I find war allies a lot more helpful on high levels, and it goes beyond "higher overall power level".
My impression is that we see more powerful offensive stacks relative to city defenders. Or maybe that's rubbish and the AI just struggles more with more units and credible threats from multiple directions. At the end of the day, I find opportunity comes knocking more often on high levels... or maybe I'm just too lazy to look as hard if I don't need to.
Early engineering for siege-based warfare works (I cut my teeth on Deity with it because I was too stubborn to trim my Liberalism path until it worked reliably) but that's something entirely different.
When hoping to snatch cities I avoid siege, wearing down defenses is what allies are there for: the idea is to cripple ally and enemy equally unless the ally is a vassal already. I may keep a few spies on hand if I think I'm going to need an edge.
Longbows will do just fine for cleanup (first strikes vs. wounded units), they're disposable and hold conquests competently. If our ally moves after our enemy, we can get away with some horse units. If our ally moves before our enemy, things become a lot less fun and we may need some specialised defenders on our own.
And I can't make a coherent guide because if I try this my play isn't coherent: "Dang, that didn't work. Oi, you aren't supposed to do that. Woohoo, AI blunder".
It's not something that I can tackle mathematically and make a clean argument for like whipping optimisation, or improvement/trait/UU analysis. Too many details play into it like map layout, AI personalities, overall diplo situation, turn order
And sometimes you can experiment on how far you can push yourself with new things before it starts breaking. 
