The decision is entirely up to you. Having an element of competitiveness is known to breed positivity and productivity. You could opt for an activity that determines a winner by way of accumulated points throughout the activity.
As is the case with any competition, a definite winner cannot be singled out in advance because individuals/teams get plenty of both subjective and objective opportunities to rack up points. Regardless of your decision — to include or exclude winning teams/individuals — always ensure that the entire activity is friendly, positive, and focused on the outcome more than anything else.
As for the second question, you may choose to include a reward for the winners. However, it’s always advisable to keep it modest. You could go for low cash prizes, gift cards, free subscriptions for paid apps/tools, or donations towards a charitable institution (or cause) in the name of the winner.