Will this box stay stable when fully expanded and loaded?

The three-tier folding design is engineered for stability when open and loaded. Test the balance with your heaviest drawer first before loading everything on the upper tiers.

Can I lock the drawers to keep them from sliding out during transport?

Locking mechanism not specified — you may want to consider aftermarket drawer catches or a bungee strap across the fronts if you're moving this between locations regularly.

Are the drawers removable for deeper storage?

Drawer removal capability not confirmed in specs. They're designed to pull out for access, but whether they lift completely free is best verified by examining one in person before committing to the purchase.