Should you version web applications

I commend your initiative. This should yield a number of benefits for the organization as you implement this. I would move the code rather than copy it. Having copies will likely result in incompatible changes which will need to be resolved.

There will be some pain as the libraries are developed and stabilized. Once that is done, the benefits will arrive. Remember that the interfaces to the library are essentially a contract with the projects working with the contract. You may have the advantage of removing old interfaces as you may be able to determine if they are used.

While the libraries are stabilizing, getting new libraries should probably be part of the code update process. You may want to schedule commits of library changes. Announcing the code being moved may help reduce conflicting changes.

The libraries should be treated as separate projects and stabilized as soon as possible. Once they are stabilized (particularly the interfaces) it should be easier to integrate changes with other projects. New libraries should work with old code. Tag stable library releases with their own release id. Try to treat the libraries as you would third party libraries.

Dates are helpful for when you need extra information about when something was happening. There is no harm to adding them to a file that already contains a version number.

However, you can not use a time stamp as the sole indicator of what version is running, therefore deciding what needs updating. Just because you released 2 updates in the last year does not mean that someone with an install date within that range is using one of those two versions. If the user gets a copy of the installer, they can hold on to it for future clean installs and expect the software to update itself instead of getting the newest version.