This relic says: "When making a ….. check to close a gate, you may return this card to the box …… to automatically pass the check."

Question: Can you take your dice roll 1st to see if you can close the gate in the normal way before committing yourself to using this relic?

No. You OR perfom the check by rolling da dice, OR activate the relic.

Right. If you opt to roll, and then fail the roll, you've gambled and lost. You'd have to wait until next turn to try again (or use the parchment).