Civilization – What’s the best way to increase the rate of research

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?

I think the best way is to manually allocate citizens in your city to work for things that generate more gold / production / food / culture over science.

For example, if you leave your citizen allocation for the "governor" (the AI), it will surely prefer a "2 production / 5 science" hex to a "2 production / 1 gold" hex, even though the latter is better for you. Remember that for manual allocation you must first open the citizen allocation menu in the city screen, and also remember you can allocate citizens in buildings, for the buildings that allow it (called specialists).

You can also leave the allocation to the AI and only direct his effort by choosing something to focus on, via the radio buttons in the manual allocation menu. Those work better when you want something specific, though, not when you want to avoid something specific.

Finally, remember that even after you have researched everything, you can continually research the repeatable "future tech" technology for extra points. It gets harder and harder to complete it every time you repeat it, so you can never have too much science if you go this route.

By the way, if you have built any academies you can now override the hex containing them with some other improvement / great person structure, that can also help if you want to redirect your efforts elsewhere.

EDIT the newest version of civ5 allows you to sell buildings - which is perfect for this case! Just sell all your research labs, universities, libraries etc. - sell all buildings that only give you science. Since you can sell only one building per city per turn, I recommend you start by selling the buildings with the highest maintenance cost.