Yes, underwriters review income, assets, and credit right before closing a file to ensure that nothing has changed in the borrower's circumstances since applying.
Yes. FHA does not cap on the number of homes a borrower can own. However, FHA does restrict borrowers from having more than 1 FHA mortgage unless you meet some exceptions.
Yes, your mortgage can be transferred to another lender after closing through a mortgage servicing transfer. This doesn't change your loan terms but shifts the company managing your payments.
Yes, but the process to remove it will depend upon your mortgage servicer. Mortgage servicers might require you to have a certain loan to value or on-time payment history.