Depends on their orbit. LEO will see drag from the atmosphere and will fall from orbit relatively quickly. Geostationary satellites will orbit basically forever.

Won’t solar winds push it out of orbit in a geologically relatively short amount of time?

Some of the Lagrange points allow for more stable orbits so would presumably be better places to look.

Depends on the orbit. Elon's roadster on its heliocentric orbit has a predicted half-life of 15 million years (1)

The moon has been on orbit around the earth for billions of year and will probably remain on orbit forever (whatever "forever" means).

(1) https://www.mdpi.com/2226-4310/5/2/57