The "spying on satellites" program is TEAL and there's precious little info on it. We know that two of its scopes (TEAL blue and TEAL amber) were used to check out Columbia's tiles to inspect for damage, which gives us an outside edge of what they can resolve from the ground. We can presume that since those scopes peeked, anything else that was up there peeked and didn't see any damage, which either tells us that there wasn't a lot of observable damage or that we didn't have anything practically close enough to really get a look. But our spysats-for-spysats program is super-spooky quiet.
Doing quick math to determine the arc second radius of a 1m satellite at geostationary orbt, it turns out someone already published the maths and did the tests. 1.5m of aperture and some fancy CCD cameras and you can get size and shape detail. A square with a length of 36 inches at GEO distance will be roughly 1/10 of an arc second; at that resolution the atmosphere will limit your ability to get an image of the object. Something 20 feet wide would have an apparent diameter of .29" which, barely, is doable with a 2 meter class scope, and perfect seeing. Thierry Legault uses a 14 inch class SCT with great optics, a kick ass camera and a ton of patience to get images of satellites in low earth orbit. There are some images of his that track GEO stuff, but GEO is a lot farther away than the ISS and the other satellites in LEO. Now I kinda want to start tracking satellites and buy a better telescope.
Then there are the rumors that they were testing a laser by tracking the space shuttles. https://en.wikipedia.org/wiki/Terra-3 Sally Ride, at a talk I went to a million years ago, was asked about it and gave a hilariously joking answer and flat out denial that they had any issues related to Soviet lasers. It was along the lines of "if we were the first crew targeted by a laser like that, we'd all brag about it an bring it up in every conversation."
Yeah. Not sure about that one. David Hoffman in Dead Hand argued that the laser complex at Sary Shagan was decades from functional. It made the news when they dazzled the shuttle, though - although I can't find any reference to it other than unreferenced wikipedia articles.