I switched branch and the invoice Add button disappeared.
The branch you switched to has no active numbering series for invoices that covers today. Ask the administrator to add one (Create and manage numbering series, Branches).
Does switching branch sign me out?
No. The page reloads and you stay signed in. Because the page reloads, save or finish what you are entering first. What does sign you out after a short warning is an administrator editing your user, your role or your branch (Who sees what: licence, role, permission, branch).
Do my documents move to the new branch?
No. Each document stays in the branch it was made in.