You choose how many cards to discard while Jaya’s second ability is resolving. You can choose to discard zero cards this way (and then draw zero cards) if you wish.

Mana produced by Jaya’s first ability can be spent among any number of instant and/or sorcery spells.

Jaya’s emblem doesn’t grant you permission to do anything with instant and sorcery cards in your graveyard except cast them. For example, you can’t cycle instant or sorcery cards with cycling from your graveyard.

Because it’s a loyalty ability, Jaya’s first ability isn’t a mana ability. It can be activated only any time you could cast a sorcery. It uses the stack and can be responded to.