Are building permits required for a kitchen remodel? I'll be relocating power outlets and some plumbing as part of it.

If you plan on moving power, plumbing, walls, or heating/air conditioning (HVAC) then a permit is required. It wouldn't be required if you are just painting, replacing cabinets, countertops, etc.

I'm interested in finishing my basement. It looks like I'll need to cut into the load-bearing wall. Is a standard contractor authorized to do that or must that be done by a structural engineer?

Generally, an engineer would be required to tell us the size of beam that would be needed to open the wall. But, there are instances where a contractor could tell you that.

Can we add a small kitchen to our basement so one of our children can use our downstairs as an apartment?

Yes, you can add a kitchen. A building permit would need to be obtained for that work.  However, if the apartment is going to be rented to a non-family member in the future, there are other requirements to make it a legal basement apartment.