If you discard a card as a cost to cast a spell or activate an ability, Surly Badgersaur's triggered ability resolves before that spell or ability but after you've chosen targets for it. If you discard a card while a spell or ability is resolving, that spell or ability finishes resolving before Surly Badgersaur's triggered ability does.

Surly Badgersaur's abilities are triggered abilities, not activated abilities. They don't allow you to discard a card whenever you want; rather, you need some other way of discarding a card, such as a cycling ability.

If the target creature is an illegal target when Surly Badgersaur's last ability tries to resolve, the ability doesn't resolve. If it's a legal target but Surly Badgersaur is no longer on the battlefield when the ability resolves, the target creature won't deal or be dealt damage.