Moving into a Courtyard Villa...
 

I mentioned to the previous owner that I was going to plant some bushes outside the kitchen window (next to the neighbor's side of house); he mentioned that I should ask the neighbor first. Is that necessary?

Read your deed restrictions. They are somewhat confusing, but I think as long as the bushes are not within 12 inches of your neighbor's house, you don't need the neighbor's permission. You need to allow your neighbor to maintain their house inside your gate, which would include access to their house for painting, inspection, pest treatment, and other maintenance.

:) https://www.districtgov.org/departme.../CourtYard.pdf

The yard is yours to use. You can plant anything you want, just keep everything from touching your neighbor's house. There is often some confusion about villas, especially by those who don't live in them. Ignore the property lines and look diagrams provided in the post by CWGUY.

the area by your kitchen window is inside your gate and yours to maintain. On the area outside your gate is your neighbors to maintain.

Exactly. It's your private yard.

Just don't let stuff grow on their house.