Is there a floor lamp that you can move around in live mode AND carry in inventory

I can't think of any, just table lamps. It would be very, very easy to add this functionality yourself to any floor lamp in the game though. Take about 3 minutes to do.

Tutorials to do the 2 additions required:
Make any object able to go into inventory.
Make any object Live Drag enabled.

The oil lamp will work that way I believe. It's not tall like the regular floor lamps, but it does look nice on the floor.

There is a couple of oil lamps that came with outdoor retreat that you should be able to carry in inventory. I think also the small Diwali candle from the holiday pack goes in inventory.

https://www.ea.com/games/the-sims/the-sims-4/pc/store/mac-pc-download-addon-the-sims-4-outdoor-retreat

https://www.ea.com/games/the-sims/the-sims-4/pc/store/mac-pc-download-addon-holiday-celebration-pack

I mainly know small lamps that can be moved in live mode and put into inventory

These are the ones I know (some have already been mentioned)
- base game oil lamp
- two portable lamps from outdoor retreat
- one portable lamp from jungle adventure
- 3 desk lamps from discover university
- 5 lava lamps from strangerville

Thank you for all the information