Hello and welcome
It sounds like you're doing all the right things. Personally I wouldn't give lettuce (or cabbage). I can't remember what the possible issues are with lettuce and cabbage, but there are lots of other veg they can have. A small bit every day as Cypher says (maybe every 3 days for a week, then every 2 days for a week, then every day).
I tend to keep broccoli, carrots and cucumber in the fridge - they all keep quite a while in the fridge. Carrots in the fridge last for weeks
Cucumber maybe a week (you can just buy half a cucumber though) and broccoli at least a week. I also sometimes get packs of baby corn or green beans. And spinach is great but doesn't keep that long in the fridge.
Just a very small piece the size of your thumbnail each day. I find we eat more veg ourselves now I get a variety for the hammy lol.
For treats I might give a small piece of walnut or brazil nut but only once a week for those. Our hammy also likes dried coconut pieces or Rosewood little hearts as treats. Pumpkin seeds always go down well