I know that indexed records can be edited if there is a blue "edit" next to the information. But, is there a way to edit information (like spelling, or even dates) if the blue "edit" sign is not there?
No. that is the only option once the collection is published after indexing - unless FamilySearch introduces some other option/reindex availability (perhaps along the lines of the new My Opportunities> Review Names or GetInvolved app). But there is not a way to request editability for records you might want to edit.