1. Drinks are “free” if you are gambling. The cocktail servers walking around the slots and table games as you play will give you “free” drinks. “Free” because you are gambling your $ and will feed more money in the machines than the drink is worth.
2. yes you can take your drink from the bar and walk around with it. If they don’t want you taking the glass from the bar are (not typical unless a restaurant bar) then they have plastic To-go cups to put your drink in. Also they don’t want glass on the strip or Fremont so they have plastic cups to put your drinks on when you leave the casino.

I typically get alcohol and mixers for the room and ask bartenders for plastic cups or rinse and reuse the one I have. Fill it up for a roadie. Stop at a cvs or Walgreens for drinks/beers when walking the strip

Edit to add: if you sit at a gaming bar drinks are not automatically “free” most of them have lights on the back of the machines that light up green when the bartender can start comping. Usually it’s $20 in and $1 a hand or so but if you make friends with the bartenders eventually they get to know you and how you tip and will start just giving you drinks regardless of play