Form 4868 gives you time to file (6 more months) but the tax is still due April 15th. So if you owe, you still need to send in a payment to avoid interest.
Yes, you continue your return and file it when complete.
Check with your state to see if you need to also file an extension with them.
"Some states including Wisconsin, Alabama and California offer automatic six-month extensions to file your state income tax return without having to file any additional forms. Other states, such as New York, will grant you a six-month extension but you must request it. For the nine states that do not impose a state income tax, you don’t even have to file an income tax return, let alone request an extension. Reach out to your state tax authority to confirm which steps you will need to follow."