1. Not necessarily. Soil bearing capacity may be calculated according to shear strength (Terzaghi among others), but is often limited by allowable settlement. You'll want to get the professional advice of a geotechnical engineer, preferably one with experience in your local area.
2-4. Yes, there is some iteration of placing columns and sizing the foundation for each column appropriately. Until you have gathered more experience, you will need to iterate to find an acceptable solution.
5. The structural engineer will work to develop a structural and foundation design which can be accomplished within the limits of the Architect's layout plan. If you cannot find a feasible solution, you will negotiate with the Architect to have them revise their plan such that columns and foundations can be placed as needed.
6. This is the same answer as 2-4. You will iterate potential solutions until you find an acceptable one. The iteration process will improve as you gain experience. After designing a few structures completely, you will understand what is likely to work and what is not.

Regarding point 1 - you should use the term "bearing capacity" when talking about shear failure i.e. when the foundation fails dramatically and plunges (to some degree)into the ground. This is the Terzaghi calc you are asking about.

The allowable bearing pressure, in my opinion, is the bearing pressure which results in settlement less than your criteria, often 25mm. This is calculated using consolidation theory (developed by Terzaghi also, but different to my first point, elastic theory, and emperical solutions (i.e. schmertmann, christian and carrier, burland and burbridg).

The below is how i typically present it.