Everyone, stay safe - Perfect and also commonly said as "Stay safe everyone"
Everyone, remain safe - Also right but less natural
Everyone, stay sheltered - This is correct but sounds like something the weatherman would say when saying to stay inside during a storm. You wouldn't typically use this in conversation.
Everyone, remain sheltered - Again, technically right but awkward
Nobody go out until the storm is over - Perfect and sounds natural - you might add "outside" depending on context
All of your example sentences work without the commas as well.