The application has been previously tested with an S60 2nd Edition device and it is Symbian Signed. Can I install the same application to other S60 devices?

There shouldn't be any problem with the installation itself. However, if the application has only been tested with one device, there's no guarantee that it will work in other devices. Therefore, retesting with other target devices is recommended. Also, be aware that applications developed for S60 1st and 2nd Editions cannot be installed to an S60 3rd Edition device as such. At minimum, it will be necessary to recompile the application with the corresponding SDK.