IP Parameter Editor not selectable

Yes, it is actually.

You can't reorganize files like this and expect anything to work correctly. You can store the files generated by an IP in a different location than in the project directory if you want (location selected when you start parameterizing the IP), but you still need access to the IP's .qip file, pointed to by the project's .qsf file (Quartus settings files) which must be in the same directory as the project file (.qpf). The .qip file is a single file that points to all the files that make up an IP. If you move stuff, the pointers in the .qip file will be wrong and the IP can't be used.

If you put your source code in a different directory (non-IP source code, that is), then you have to make sure the project is pointing to the source code (manually edit .qsf or point to files in Add/Remove Files from Project in Quartus Project menu). I'm not sure what "constraint files" you are referring to (I/O?) but you can source other files from the main .qsf file with the Tcl "source" command.

There are ways of doing what you suggest, but you have to make a number of adjustments in assignments, not just move files around, to make everything still work.