Can I put an object thats still building in the ship so I can work the area better?

No, items that are still building cannot be stored.

Afraid not. Items under construction cannot go into storage.

No you can't. You can only store completed decorations on the ship. You could however temporarily move completed objects from other parts of your island onto the ship to make room for the object under construction for the time being so that you can work more freely.

No, you have to wait until construction is complete in order to store it. However, you can move/drag them to any empty spot the same size and get them out of your way.