Yes h1b be can be approved for people who are out of status. But they have to go back for visa stamping. Its when it gets interesting. Since they have been out of status they will be banned for entering USA depending on the time they have spent as an undocumented alien. Meaning if someone overstayed for 6 months they will ban for 3 years and if its more than a year it is 10 years. There is a waiver for that but it is almost impossible to get such waivers.

And yes you have to be in a some status (f1 / h1 / any other status) to start GC processing. if you don't have any status you are out of lock.

TPS doesn't provide a way to adjust to any other status.