How does satellite tracking work? Does a country like Russia or US track every single one of its satellite and each others' satellites?

Imagine if Russia or China just launched a satellite that showed the capability to grab and take apart & examine American satellites while in orbit.

They can clearly destroy them but having the ability to examine them would be a bit of a game-changer, since it is assumed satellite with latest super-secret techs are safe after the launch?

Do some fancy satellites have proximity sensors and self-destruct mechanisms...?

I know Russians have built one of their manned spy sats ( Almaz?) with an anti-aircraft gun designed to be used if American satellites would approach it.