You really have to take the baby's lead, rather than thinking about what should happen at a particular time. They create their on schedule based on their feeding cycle.
At 8 weeks their gut is still small so intake won''t be massive and therefore sleep cycles are short. I found with an alert baby that the routine went like this: wake up, feed, change nappy, some sort of interaction like singing, pop in baby bouncer for play, rest of feed, back in bouncer or nod off in arms.
Like your baby, ours didn't sleep for very long. I just got out and about. The crying half way through the walk is familiar as hunger - I couldn't get very far without another feed, especially in the morning. I learnt not to fight it: just stop and feed. Then carry on. go by their schedule.
Mine was gassy too and we found infacol very useful. You pop some drops in and it somehow makes the little bubbles of wind in baby's tummy join together and become one bigger fart/burp. You can buy it over the counter and use it for each bottle.
Stupid question, but are you giving the correct amount of formula? You might want to try a formula for hungry babies so she gets more carbs per mouthful, although I've heard some babies don't like it so much so you'd have to try it and see.
I agree that you need to stop thinking about a 7 o clock bedtime and personally I think spending a couple of hours in a quiet environment might be a waste of time. We just kept baby with us downstairs until we all went up to bed. We had a cot next to the bed.
Regarding night time feeding, babies naturally feed a lot in night because that's when mum produces a hormone that tells her how much breastmilk to make. I know you're not breast feeding, but baby will still have this natural instinct. The night feeds start to reduce at 3/4 months - basically when the baby's gut is bigger and can hold more milk. The gassiness reduces at this time too.
So all in all, just go with the flow, follow baby's lead, get a bouncer, take them to baby club in the morning, get a kip midday when they have a bigger sleep, get some infacol and use it regularly, throw out any schedule and follow baby's cues, tell your wife and yourself you are doing amamzingly well and baby will start to sleep in bigger, more regular chunks when their belly is a bit bigger.