I have used a piece of lathing wire (Home Depot $7.00 for a 2 foot by 8ft) and made a cover for the drain for several people. One of those people, every time it rained and the drain blocked up, the water would fill in the cellar stair well as much as 4 feet. It broke out the storm door and leaked on into the basement. They had a flodding problem similar to yours, every time it rained. You might find a smaller piece of similar mesh openings somewhere. Don't use window screen, it'll plug up fast. Take a small bowl and shape the mesh so it will sit a bit higher than the drain. That way, when the debris collects around the base causing the water to rise, the water will then enter the drain.

As for the cats, they hate orange peel and loath banana peels. Use them for about a week and they may quit coming around. Don't leave them there too long or you'll get insects and other vermine trying to get in the door. You could also sprinkle some moth crystals in the area, they work very well.

If possible, this area should be finished to a roofed enclosure to protect the drain from the monsoons and rainstorms.
Temporally, your treed neighbor could remove the leaves when you are gone on your European vacation...
Or, use this money for the attached entrance covering.