Can I conquer Hall of Legends and, if I already used Rek'Sai's legend ability, pay the cost for conquering the battle file and then use Rek'Sai's legend ability again?

No, this does not work. If your Rek'Sai (Void Burrower) is already exhausted when you conquer Hall of Legends, its "When you conquer" trigger condition is checked at that moment — and since it must be ready to pay the exhaust cost, the trigger never enters the Chain at all. Hall of Legends can ready it afterward, but the opportunity to trigger has already passed. (FAQ #11259, #11354)

The one scenario where these two work together is the reverse order: if Rek'Sai is ready when you conquer Hall of Legends, both triggers fire simultaneously. You order Rek'Sai's ability to resolve first (exhaust, reveal, play a card), then Hall of Legends resolves — letting you pay 1 to ready Rek'Sai for the next conquer.