Intrepid Paleontologist must be under your control in order to start casting a Dinosaur creature spell using its last ability, but it doesn't need to stay under your control throughout that process or be under your control as the spell resolves. If, for example, you sacrifice Intrepid Paleontologist to pay an additional cost of the spell or it's destroyed in response, the spell will still resolve normally.

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.

Multiple finality counters on a single permanent are redundant.

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.

Finality counters work on any permanent, not only creatures. If a permanent with a finality counter on it would be put into a graveyard from the battlefield, exile it instead.

You must still pay all costs and follow any timing restrictions and permissions for Dinosaur creature spells you cast from exile this way. Normally, you'll be able to cast them only during your main phase while the stack is empty.