First of all, lets differentiate between:
1) The shortcut: F3 versus ^L
2) The actual functionality
1) The shortcut: This is configurable.
Both TSynEdit (as a component, in ObjectInspector) and the IDE offer full configuration of keyboard shortcuts.
2)
Search again exists in the IDE.
SynEdit itself does not have a search nor search-again command. That is, because SynEdit does not include any dialogue to do a search. Such a dialogue must always be provided by the application using SynEdit.
SynEdit offer methods, that can be called to do the actual search.
Back to previous position
Again the IDE has "history jumps" (back and forward). Assign the keys you want.
I do have to check, if a history point is created, when a search is started [A]. And also if or if not, in case of a "Search again"
[A] Actually, it would be created when jumping to the found search result. And it will only be created, if that is at least a few lines from the previous location.
You could record editor-macros for "set history point", "search" => that would probably always set one (not sure if it needs to be a bit away from the previous history point). You could then assign that macro to your key-combo.