That's not true; it also measures government spending (e.g. on building infrastructure), which cannot be taxed because it makes no sense for the government to tax its own spending. Gross Private Product is the GDP equivalent that only includes taxable (private) activity.
I am pretty sure government spending is taxed in most countries. Both companies and employees pay taxes. Maybe you meant government revenue? (but then that’s the tax itself!)
Some government spending is immune to some taxes which would otherwise be paid on similar private transactions (which taxes and transactions this applies to are different in different jurisdictions.)
Yep, government employees still pay income tax, and government contractors pay taxes on profits.