How do you cope with unhappiness in Civilization 5?

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?

Have you considered leaving cities as puppets for a while?

Unfortunately, you will have take the cities with your military rather than taking them in a trade for peace. However, if a Civ is giving you a dozen cities for peace, chances are you have them beat and taking the cities will not be a problem. And for your trouble you get:

• Puppet cities do not cause nearly as bad of unhappiness as an annexed city.
• Since you have to take the cities with units, you'll get the cities a lot slower, and spread out the impact on your happiness. Thus more time to react.
• If you really want to control each city, you can annex them later. However, you can spread out the effect even more by annexing them one at a time, building a courthouse, then annexing the next AFTER the first is happy and useful.

Though already answered, I am going to throw in my own two cents because managing happiness is one of the biggest challenges in Civ. V.

Step 1) Realize that unhappiness is inevitable. Cities will grow until unhappiness stunts their growth. Therefore, always be on the look out for ways to increase happiness.

Step 2) Luxuries are a major source of Happiness, especially during the early to mid-game. Be sure to mine, quarry, or build a plantation on every luxury within your empire's borders, and choose new city locations based on their proximity to new luxuries.

Step 3) Explore and meet your neighbors. Each individual type of luxury only gives your civilization the same +5 happiness, i.e. having 2 sources of silk does not give anymore Happiness than just having 1 source. However, if your neighbors have excess luxuries of their own, you can swap different luxuries giving each nation an extra +5 happiness. Word of warning, if the luxury listed has a one in the parentheses like this "(1)", then it is the nation's last bit of that particular luxury. AI civs will be very loathe to part with it and their price will go up.

Also, I have never seen the AI iniate an offer to exchange luxuries. The player needs to routinely check the diplomacy menu to see if any nation's have excess luxuries you might trade. Sometimes, you can pay several hundred gold in lieu of offerring a luxury of your own.

Step 4) Save your pennies to bribe City-States. Bribing city states have a lot of benefits, including extra food and culture. Now, if that city state is an ally, then also share their strategic resources and luxuries with you. So, browse the Diplomacy menu to see which City-States have luxuries you do not. This is a great way to pick up extra happiness.

Step 5) Build Coliseums, Circuses, and Theaters. These buildings take a long time to build, but are ultimately one of the few ways to garnish happiness on a widespread level. I mention these buildings for their economy, they give the best amount of happiness for the gpt maintenance.

Step 6) Control expansion. If your Happiness is already at 2 or 3, there is no room for another city until Happiness goes up (an exception might be made for a city that adds a new luxury to your empire). This also means do not acquire enemy cities when your happiness is already low. Since there is a Happiness ceiling on expansion, a player should only acquire enemy cities that are worth keeping in the long run. Otherwise, they should just be razed without ceremony. If necessary, a settler can always be dispatched to reclaim the location under more advantageous circumstances.

Step 7) Annex cities 1 at a time, and immediately build a court house. Puppet cities can be annoying with their tendency to build Barracks and Arsenals when they cannot build military units. Yet, an Annexed city will provide more than double than unhappiness as a normal city (at least 6 unhappiness plus population*1.2). Keep in mind that is on top of unhappiness it creates as a normal city. There are a lot of benefits to annexation, the process just needs to be planned out so as to not stifle the nation.

I did not discuss any nation specific buildings or strategies, but there are out there. India in particular gets a significant reduction to unhappiness caused by population.