make sure they have some sort of rewards attached (Cashback, travel)
As many as you please.
I have 6 as well but one of the companies that I already have a card with keeps contacting me about a different card that I'm pre-approved for.I have about 6. I never close my cards. But I do lock the ones I no longer use. I'll purchase something small and pay it off just to keep it somewhat active though.
Only actively use 2. Both are cashbacks.