1) I think any substantial soft sided insulated lunch bag/box will do. Some flat ice packs like these come in handy.

2) No thermos suggestions, sorry.

3) I just bought a Uniqlo ultralight down jacket and I love it. It comes with a little pouch to stuff your coat into. Humans have spent the last sixty years trying to invent a substance that works as well as down and they still haven't.

4) While I personally love my Tom Bihn backpack, I recommend you check out this thread on /r/buyitforlife to find the best bag for your needs. Actually, there's discussions for jackets and lady gear there too.

5) Up north around Seattle, we have ORCA instead of BART. I have this ridiculous little wallet, but I know they come in a bunch of different styles. It's 3/4 of an inch longer and an inch taller than my credit/debit/ID/ORCA cards are. It's nice and thin and the RFID goes right through it.