1) No, an organization does not need a building. Buildings assume you have non-trained workers that can at least operate those rooms. But you could also say that your organitation's workers are using that room and sum the total of capital earned by them. More can be found on the topic Combining Rooms and Teams.
An organization may or may not have a base of operations.
For a room, the Earnings amount already subtracts the cost of having unskilled employees to do the basic work for you. For example, the Earnings listed for having a Bar already account for the wages of a bartender and servers.
2) Nothing prevents a building from being used by anyone, but any physical limitation is left to your GM. How big is the room? How many people fit into that room? Keep in mind that for a building to produce capital, it must be worked for 8 hours a day (at least), with any other period being free to be used by you or anyone you allow to use it.
3) You can combine them, or not, the choice is up to you. Regardless of your choice, you will make a single roll to earn capital per capital being produced (there is a error in the rules here that i will mention later), once for your buildings and once for your organizations. That means that you can have a Team of Acolytes earning Influence and their temple earning Magic, or they could combine and earn GP. You would roll once for Influence and once for Magic, or only once for GP.
You can read more about this on the topic about Income Phase.
4) No, they only generate GP if you decide to generate GP. What is true is that nearly all buildings CAN generate GP (i believe all on the book do), but that doesn't mean they generate two types of capital at once (say, influence and gp), unless your split the bonuses properly, as described on the rules.
This is explained on the example given:
Example: The Inn example in the sidebar Construction Examples has a Bar, a Common Room, a Kitchen, a Lodging, and a Stall. The Bar gives gp or Influence +10; the Common Room gives gp or Influence +7; the Kitchen gives gp or Goods +4; the Lodging gives gp +12; and the Stall gives gp, Goods, or Labor +8. All of those rooms can earn gp, so if you want to earn gp, just add all the room bonuses together (total +41) and make a skilled work check for the Inn to see how much gp you earn. If you wanted to use the Bar's bonus to contribute to generating Influence and use the rest of the rooms' bonuses on gp, you'd attempt one skilled work check for Influence with a +10 modifier (the Bar's bonus) and another skilled work check for gp with a +31 modifier (the total bonuses from the Common Room, Lodging, and Stall).
There is a contradiction on the rules during the Income Phase, which says you make a check per building or organization, but later on the book it says you make a single check per capital. This has been discussed a lot on paizo's forum (this thread specially), and since it will see table variance, i suggest that you take a look at the thread i linked and take your conclusions.
If you have multiple buildings or organizations in a settlement and they can generate the same kind of capital, you don't have to roll for them separately—you may add all their capital modifiers together and attempt one check for that kind of capital.
If you do allow multiple checks per building/organization, you will earn more capital, but it will slow down the game considerably. If you make a single roll, the difference in capital is small, but lower, but you will also make the game much faster.
What i suggest is that you allow multiple checks if different buildings are producing different capitals (a bar producing influence and a temple producing gp), but if they all produce the same type of capital, add all their bonus together and make a single check.
Just a heads up, Chemlak is possibily the community member that understands the most about the downtime system. I cannot recommend his posts enough, because other than him, you could only ask Jason Nelson yourself (the creator of the system).