I would collect the rocks and clean them up. You can plant a tree or large shrubs. I see a lot of trees in some of the photos. If a tree can grow on a city street, a tree could grow there. You can then use the rocks as a mulch to help keep the ground moist or use them for something else. I would probably plant an evergreen. You don't want something that has branches that go over the court.

I would expect that trees' roots would find water within a year or two of planting. You would need to water for the first year or two after planting. You should be able to find a native ground cover that would look better. What zone and nearest city to you? Sun/Shade?

My bet would be the cheapest thing would be to pin some of the ivy onto the rocks and let it root.

If I‘m seeing those hills right you are in a drier climate. If you don't get a lot of plant litter in that area you can pull out all the rocks, remove weeds and put down landscape fabric with rocks on top. Otherwise just adding more rocks should help, the thicker the rock mulch the less weeds.

I wouldn’t add trees close to sidewalks, the risk of the tree roots raising the concrete is too great.

With rocks you could look into a blow torch and just burn the weeds.

It’s called ‘flame weeding’ and done all the time with the proper precautions.

That would certainly sterilize any weed seeds! I would hire a pro for flame weeding that knows what they're doing and make sure they do the cleared patches for those seeds.

Nothing like inadvertently sparking an out of control grass fire to make you the neighborhood's new favorite person! :-)