Civilization – Do the Colosseum and Theatre need separate population for their benefits

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?

Here is a list of all the effects from happiness level that I could find. To answer your specific question, most of the effects care only if your civilization is happy or unhappy, by one :) or :(. (I believe zero :) is fine.) But there are three effects that care about the amount your civilization is happy or unhappy:

• VERY unhappy occurs when you hit -10 happiness. (Thank you @Oak)
• Golden Age points accumulate (or deplete) equal to the amount your civilization is happy (or unhappy)
• Piety->Mandate of Heaven uses your exact excess happyness to calculate how much culture you get.

Full list of effects:

Happy

• Growth: Cities grow normally.
• Golden Ages: If you have excess happiness, each turn it accumulates into Golden Age points, and when you have enough, your civilization enters a Golden Age. After the golden age the process starts again, although I believe you need more Golden Age points for each subsequent Golden Age.

Unhappy

• Growth: Cities grow at 1/4 speed when your population is unhappy.
• Golden Ages: The amout your civilization is unhappy is subtracted from the Golden Age points.

VERY unhappy

• Growth: Cities do not grow.
• Units: Settlers may not be trained.
• Production: Cities generate -50% production.
• Unit are less effective in combat. They get -33% combat strength.

VERY VERY unhappy (-20 or lower)

• Rebels will spawn in your border and attack your city

Social Policies

• Piety->Mandate of Heaven allows you to add 50% of your excess happiness to your culture each turn.
• Rationalism->Sovereignty boosts science +15% when your empire is happy.
• There are numerous social policies that affect the happiness level, but these are the ones that get effect from the happiness level.

Sources:

• General info: mousing over the happiness listing on the top bar, left side, during various happy states.
• Golden age info: mousing over the golden age display on the top bar, left side, during various happy states.
• Combat info: Comparing happy (normal) combat strength to VERY unhappy combat strength, by mousing over an enemy unit. The VERY unhappy combat modifier is clearly listed in the calculations.
• Production info: mousing over a city's production listing in the top left when VERY unhappy.
• Social Policy info: The social policy screen, accessible by clicking on the culture listing on the top bar, left side.