When I was in hospital, I spend 15 minutes for both side each time. When my baby was around week 2 or 3, 10 minutes is enough because he learns how to suck better and your milk flow is faster too. Drinking from bottle is faster because he uses less effort to get the milk out. Your situation sounds similar to mine so dont't worry too much! I. Your baby falls asleep and do not wake up when you put him down, I think he is full. My baby wakes up when I put him down sometimes and it frustrates mr because I have to feed him again！
At first I mixed formula with breastfeeding. Then I realize he drink faster and less fussy with the bottle, so I expressed my milk for him.