missmissy,
Fridge-to-go is a portable cooler bag with built in cool pads. Very good when you return to work and still pumping:
http://www.fridge-to-go.com.sg/
Breastfeeding is tough for first two months but it gets easier over time. Depending on individuals' tolerance of pain, some find latching painful. Cos your bb is less than a month old, it could be your nipple still not 'seasoned' yet. Latch her on one breast till she pushes the nipple away herself and falls back to sleep with the 'drunken' look. You mention your baby is gaining weight. That's a sign she's getting enough milk. Other signs to watch for are wet diapers and poo. Make sure she has wet diapers and poo daily.
Newborns can be quite tough to bf cos they fall asleep halfway. Tickle the cheek, or if needed, change the diaper to wake her up. If all else fails, just be prepared to latch again an hour later. Once they get enough practice, they will be able to latch shorter and fill up faster. My baby now 3.5months old just needs 15-20mins and he's done.
With regards to your questions:
1. She's still newborn. Need lots of practice and patience on your part for her latching to improve.
2. Some babies get nipple confusion but some don't. My baby is bottle fed with express milk during the first month of my confinement cos aunty is around to help. i took the chance to get my full rest. I concentrated on pumping 3hourly even at night to establish my supply. I re-latched him from 2nd month onwards and maintained one bottle feed per day cos I want him to get used to bottle feeding when I return to work.
3. For my case, with my first baby I was on exclusive pumping and my supply was good. Because from the start, I was pumping beyond her demand and i maintained it all the way. You have to be disciplined and pump 3hourly even at night. For this second baby, once I re-latched him in 2nd month, my supply dropped to meet his demand. I learn now that pumping too much can be problem too cos your freezer runs out of space and you get engorged very often. My first bfing I had to endure lots of plugged ducts cos i didn't latch her. This 2nd baby, I latch him at night now that I'm back to work and so far, no major plugged ducts.
4. Initially I also tot that after pumping the breast will not generate anymore milk. But I realise that it still can! After I pump finish, I latch my baby and I can still get letdown so means there's still milk. The baby can strip more milk than a pump. A pump cannot empty your breasts as well as baby.
Based on my two experiences of exclusive pumping vs pump/latch, latching is the best. It's convenient and prevents plugged ducts. I can go out shopping without bringing milk. Just find a nursing room. Just that latching needs ALOT of patience and practice. Once you get the hang of it, it's very easy. Over time, you will understand your baby's pattern and know when she is asking for milk and when she simply just wants to suckle for comfort.
I'm back to work now and still pumping 3hourly at work and latching him at night 3hourly (yes, he's still not sleeping thru). I use the lying down position to latch him at night so i can sleep at the same time. Hope the sharing of my experience helps.