You can continue using PA 5 (or PA 5.1) indefinitely in order to typeset any projects which have remained in Paratext 7, and have not migrated to a newer version of Paratext. Please keep in mind that the Send/Receive servers for Paratext 7 have been shut down and therefore getting project updates over the Internet is no longer possible. If any projects remaining in Paratext 7 are migrated to Paratext 8 you will need to begin using PA 6 or newer to typeset the project.
In terms of typesetting features, PA 5.1 and PA 6.0 are identical.
The major difference is that PA 5.1 works only with Paratext 7 projects, while PA 6 works only with Paratext 8 or newer projects. Also keep in mind that PA 6 requires separate registration.
The answer to this question depends on the project that you are about to typeset. If the project is in Paratext 8 or newer then you need to migrate to PA 6. If the project remains in Paratext 7, then you need to stay with PA 5 and should consider upgrading to PA 5.1. Note that PA 5 and PA 6 can safely coexist on your computer and will allow you to typeset both Paratext 7 and Paratext 8 or newer projects respectively.
Yes. Simple style templates (.isty) and template bundles (.tb) created in PA 5 can be used in PA 6. In the case of template bundles be sure that you use the same or later version of Adobe InDesign that the bundle was created with. In PA 5 these files are stored in your My Paratext Projects folder. You will need to copy them to _PA_StylesAndTemplates in your My Paratext 8 Projects folder in order to use them in PA 6.
Yes, you can safely keep Paratext 7 and Publishing Assistant 5, together with Paratext 8 or 9 and Publishing Assistant 6, on the same system.