"Save Configuration" in Drupal 7 Module not visible
Sounds very much like a custom module is hiding the module save button to stop people adding/changing things. I've had the misfortune of inheriting sites with these sort of "security features" before too. 

If you do have any custom modules installed. the simplest thing you can do to test is get Drush installed, run drush dis CUSTOM_MODULE and check to see if the button comes back. Then you'll know if it's the custom module's fault at least. You can also disable them manually in the system table but you should only do that if you really know what you're doing.

The upshot of that is you can also use Drush to install the other modules outside of the UI, so even if that's not the problem you'll still be able to install new modules.

Also, to find out if a change to core files is responsible for the problem, grab the Hacked module and run a report. If you install the Diff module along with it you'll get a side-by-side diff of exactly what has changed in core/contrib modules.