Is investing in a bank that offers a variable daily or monthly profit rate (around 10%), or in an Islamic investment company that offers a variable annual profit rate (around 16%), considered permissible, or does it fall under usury (riba)?

There is no objection to dealing with Islamic banks that have a reliable Shariah supervisory board, and investing money with them. The bank mentioning an expected profit margin without guaranteeing it, and the matter being subject to profit and loss, does not affect the validity of the transaction; this is due to the banks' caution and their study of the expected profit margin, and they are not obligated to pay a fixed amount to the investor, unlike usurious banks.