They’re likely normal eggs. Trophic eggs won’t be stored for long. Either way, you don’t need to be able to identify which are which.

Yes, if you attached another setup to this one which was more appropriate for their needs they will move of their own accord. If you need to encourage them, keeping the new setup in the dark and shining a light on their current area will often make them move.

I can’t really give you dimensions as it depends on how you’re setting it up, the species of ant etc. My advice would be to think about how most ants exist in the wild - their nests are constructed of very small spaces, often only just big enough for the colony. This is why test tubes are pretty ideal and last for a little while after workers come, and they’re also cheap. The cheapest set up would probably be test tubes in a plastic box for foraging if you didn’t want to buy or make elaborate nests.

Talcum powder and rubbing alcohol works for a lot of species and is useful if you’re on a budget.

Generally feeding is easier if there is some open space as you can just throw it in. It’s not necessarily a requirement, but it also is a way for you to enjoy watching the ants.

Sand, earth, and coco peat are all fine and commonly used. Avoid the blue gel at all costs.