Can each variant have its own weight, size and shipping cost?

Yes. In the Variants card, each row has a Details button.

1. Click Details ▸ on a variant row.
2. Fill in that variant's Weight (choose Kg, gms or lbs), Dimensions (cm) (L, W, H) and Shipping cost.
3. Repeat for the other variants.
4. Click Save.

Rows that have details filled in show ● Details, so you can see which variants are done.

For a variable product, set these values per variant here. The separate Shipping card is only shown for products without variants.