Once Rowdy Crew's triggered ability begins to resolve, no player may take other actions until it's done. Notably, you can't discard or cast any of the cards you draw to try to rig the results of the random discard.

The card types that can appear on the discarded cards are artifact, battle, creature, enchantment, instant, kindred, land, planeswalker, and sorcery. Legendary is a supertype, not a card type.

The discarded cards just need to share one card type. For example, Rowdy Crew will get two +1/+1 counters if you discard an artifact creature and an enchantment creature.

Rowdy Crew doesn't get more than two +1/+1 counters if the discarded cards happen to share more than one card type.