Is the resource obtained from the battlefield Threshhold of the Gray persist throughout the whole turn? Also is this energy only to be used during the showdown?

Yes. It stays in your Rune Pool for the rest of the turn, and there's no restriction forcing you to spend it during the showdown.

Threshold of the Gray reads "When combat starts here, the attacker and defender each [Add] [1]." All abilities with the action "Add" simply add Energy/Power to the Rune Pool (Rule 168), and a Rune Pool empties at the start of each player's Main Phase and at the end of each player's turn (Rule 167). Since combat happens during the Main Phase, that 1 Energy is still there afterward and lasts until end of turn — for both the attacker and the defender.

Unlike something like Diana, Scorn of the Moon (which explicitly says "Spend this Energy only during showdowns"), this card has no such wording, so the Energy can be spent on anything you can pay for later in the turn. The only caveat is that if you don't use it, it's lost at end of turn.