Can a tool cupboard be moved?

There is no way to move a tool cupboard as of now. The way a tool cupboard works is that for every part of your 'base' a part of its total costs needs to be placed in your cupboard as upkeep. Being able to move your tool cupboard whilst it has resources in it would mean a movable inventory in the game which is not possible. You can (as long as you have building privilege place a second tool cupboard in your base. This one won't take effect however since the game only tracks the first tool cupboard placed. It's also not possible to place a second one before destroying the first one since the game doesn't recognize the second one as placed since at the time of placing there was already a tool cupboard there. The only exception of this being that you place a second tool cupboard outside the area of the first one. This would treat both as a big tool cupboard and share the costs as long as both tool cupboards are connected via foundations.

Alright, that was a massive brain dump. So here is a TLDR;

Can I move my tool cupboard?

No, you cannot.

Can I place a second tool cupboard so my base will not decay?

No, you cannot (usually).

Also, someone please correct me if I'm wrong. I recall that you cant place a tool cupboard on anything above the ground floor since doing that wouldn't protect floors below it. This obviously is to prevent massive towers with their tool cupboards 15 stories up encased in reinforced metal.

In all honesty though. As long as no other players are around there really is no reason not to smash your tool cupboard to pieces and place a new one at the new location.