Are intra state and out of state ONLINE savings/checking/CDs/money market accounts FDIC insured same as local brick & mortar bank accounts?
Usually they are, but of course you should check the registration number the bank shows against the FDIC website to make sure you don't fall for a fake.
In case the account is offered by a credit union rather than a bank, they will have insurance through NCUA instead of FDIC, verify NCUA charters here. This applies to both online and brick-and-mortar credit unions.
Money market accounts in particular may be provided by yet another type of financial firm: a brokerage. Those are insured through SIPC and there are significant differences in what protection is provided vs FDIC and NCUA insurances. Note that many brokerages provide a "sweep" service, such that when not actively involved in a trading transaction, money is actually held in a bank account where FDIC applies. The particular broker should be able to tell you which bank holds the sweep accounts.
They can be. You will have to check the specific bank's terms to make sure, but there is nothing preventing them from being FDIC insured.