Let me go through your questions one at a time.

1.) What happens to buildings that exist in both cities?

When merging two cities you can pay either with Influence or Gold to harmonize their infrastructure. Think of Influence as political clout to convince the mayor/ruler of that city to invest in additional infrastructure to get the second city up to date. Gold is simply hard cash that you pay yourself to build that infrastructure.

You already receive a discount when the second city finishes/posses a building that the first one has - the merge price drops, sometimes by a couple of thousand units. The more expensive the building the bigger the impact it will have on the price. You are not losing any buildings in the merge, you just build them automatically in the new territories by paying the merge price.

2.) How are these permanent-bonus-giving "mini-projects" (Festival, ...) treated?
They are called Repeatables or Public ceremonies ingame. The effect is added on the tile of Main plaza itself or if you merge cities I guess it should remain on your administrative centre. It will be difficult to abuse them since the bonus they add is not that significant. You will need to let a small city build only Ceremonies for 100-200 turns to be able to really make an impact.

3.) How are the costs for absorbing cities determined?
They are not random - the game takes into account population, distance and most importantly infrastructure. As in question 1, if you build a Nuclear Power plant in city 2 you will get a bigger drop in price than compared to building a Granary. The price of the missing buildings in production is directly added to your merge price. I am still to find the exact formula.

The main way to drop the price is to simply build up that infrastructure - I would advise by buying it with gold.

The duplicates are already considered in the merge cost (and thus reduce it). There is no need for an additional bonus, I would say they need probably to add a tool tip of how the cost is calculated and what adds to it (i.e. missing buildings).

On the topic of Ceremonies I will test it later today and provide screenshots. My theory is that after merging the cities the new administrative centre (which was previously a Main plaza) will keep the extra yields that it had from Ceremonies.
I was referring to the fact that +5 food (from Feast for example) is not that much and you will need a lot of time building only Repeatables for that strategy to actually be viable. Smaller cities do not have cheaper ceremonies but a larger number of cities can build more ceremonies simultaneously.

About constructing or buying your missing infrastructure the answer is - it depends. If you use Influence for land acquisition then buying the buildings is overly better for you. It is both faster and gives you immediate benefits in the second city for the next few turns until you manage to lower the merge cost enough to unite the two cities. Even if the ratio of the buy price of a building compared to paying for the same building with the merge cost is 1:1, you are still better buying each building separately to get its effects sooner on the territories and population of the second city. This is especially important for gold-generating buildings, as you want to both improve your gold income and lower the merge cost at same time.

If you are playing a merchant empire then Inherited lands can also be great for you as you will have plenty of gold to spend. What I do is leave the cheaper buildings in the build queue (those that can be completed within a turn or two) and then buy the expensive ones. This gives your second city something to do and speeds up the merging process.

It is up to you how you want to set up your Civics and economy for a quicker integration of your cities. I do not have the exact formula of the merge cost and I can not be certain what is the effect of population but from my experience it is very low. Infrastructure should consist about 80% of the total cost of the merge.

Keep in mind that sometimes it is better to have multiple cities for geographic or other reasons. Developing then separately can also be beneficial.

Yutterh wrote:

So you're saying it is cheaper to merge the cities when they have duplicates? 

Yes