I installed extruded polystyrene on our basement (above ground). I need something to fill in large indentations in the foam. I've tried using spackling as recommended by a building supply but understand that it's not suitable for exterior applications. I've also considered water putty but have been told that it is not useable in exterior applications. Also, I need to find a tape that would be suitable for the seams. I was planning on using Gorilla tape but it's rather thick. Any suggestions for these 2 problems would be greatly appreciated!

If I understand this right... this is the OUTSIDE of the structure, all ABOVE ground...

Well, couple things here, #1 Polystyrene isn't UV stabile, and its not intended to be the siding.

Typically there will be a Stucco or other external siding material over this insulation material.

I am aware that it isn't appropriate for external siding. I previously painted it which was fine until the cat made a bit more than paw prints. At any rate, I'm trying to repair the damage as well as fill in other indentations. I'm thinking of using lattice (privacy) which will cover the scratches but would like to know what will last on exterior. I will then paint it. I would also like to know how to treat the seams. Thank you!

Go to Sherwin Williams paint guys. They have a product that is a concrete overlay that adheres to polystyrene. It is designed for concrete, but it will work for exterior and be waterproof. I don't know if your Sherwin Williams sells it, but they do here in Beloit WI. To cover the seams, you would use fiberglass mesh. Put a thin layer of the overlay on the polystyrene, put the mesh over the seam like you would on drywall. Let dry. Then put another thin layer over the seams and the polystyrene. I've done this on houses a hundred times. It works and will last. If you need more info, let me know. I can describe how to do it if they can't help you at Sherwin Williams. You have to put this stuff on either with a plaster sprayer, or a trowel. Either one.