Finality counters aren’t keyword counters, and a finality counter doesn’t give any abilities to the permanent it’s on. If that permanent loses its abilities and then would go to a graveyard, it will still be exiled instead.

Finality counters don’t stop permanents from going to zones other than the graveyard from the battlefield. For example, if a permanent with a finality counter on it would be put into its owner’s hand from the battlefield, it does so normally.

If a permanent with a finality counter on it would be put into a graveyard from the battlefield, exile it instead.

Multiple finality counters on a single permanent are redundant.

If Hundred-Battle Veteran is countered after being cast from your graveyard, it will return to your graveyard. It may be cast this way again. (That doesn’t count as one of the hundred battles.)

Once you begin casting a spell from your graveyard, it immediately moves to the stack. Players can’t take any actions until you’ve finished casting the spell.

You still pay all costs and obey timing restrictions when casting Hundred-Battle Veteran from your graveyard with the permission from its last ability.