Civilization “doesn’t want anything to do with me right now”

In war the AI sometimes gives up and hands over like 12 cities at once for peace (which I'd be stupid not to take)

I think your conclusion here may be false. If those cities are going to drive your happiness into the ground, it may not be in your best interests to take all of them at once. There is no magical wellspring of happiness that will allow you to absorb an unlimited number of enemy cities into your empire without repercussions.

Use the following methods to increase happiness:

• Build courthouses in captured cities. This negates unhappiness from occupation.
• Get as many luxury resources as possible (by building the appropriate improvements on their tiles or through city-state allies). Look for resources just outside your borders and consider purchasing those tiles. Plan new city locations with an eye toward where you can get the most luxury resources.
• Build happiness-producing structures: Circus, Colosseum, Theatre, Stadium, and Burial Tomb if you're Egyptian.
• Build happiness-producing wonders like Hanging Gardens and Eiffel Tower.
• Get social policies (there are many) that work with your existing empire to increase happiness. Look at how they generate happiness, and find the ones that match your style of rule. Don't try to change your whole empire just to eke a little more happiness out of a social policy.

In addition, use puppet cities to your advantage, and don't overextend yourself. If taking all those cities is going to cripple you, it may not be worth it. Is a vast, dysfunctional empire better off than a slightly smaller, functional one?

First of all, you can just sell them if you don't need them. Not only do you immediately get some gold, you also decrease the number of units you own, and units cost gold each turn.

Secondly, connecting a city to the capital by railroads greatly increases a city's production, so if you can afford it it's recommended.

Finally, you can replace improvements you don't need / want with other improvements. For instance, for cities that are big enough it's probably better to replace farms with mines or trading posts.

You can set your workers to "automate", and they will automatically build roads, replace them with railroads if available, and will replace improvements with other improvements the AI thinks you need more.