You may cast a card with flashback from your graveyard. If you do, you pay its flashback cost rather than paying its mana cost. Flashback doesn't change when you can cast the spell, so a sorcery with flashback can still only be cast during your main phase. Also, casting a spell with flashback doesn't change its mana value. Mana value is always based on a spell's mana cost, even if you pay an alternative cost to cast it.

Prowess is a triggered ability that gives the creature that has it +1/+1 until end of turn whenever you cast a noncreature spell. Any spell you cast that isn't a creature will cause prowess to trigger. Remember that lands aren't spells, so playing a land won't cause prowess to trigger. Casting a spell that's a creature and also another type (such as an artifact creature) is still casting a creature spell, so prowess won't trigger in that case either.

Planeswalkers are cast as spells, just like any other nonland cards. There's a number in the lower-right corner of each planeswalker card. That's its loyalty. It enters with that many loyalty counters on it, whether you cast the planeswalker as a spell or if it entered some other way. It needs those loyalty counters to stay by your side. If a planeswalker ever has no loyalty counters on it, it's put into the graveyard.