Fresh Start with selected table imports?

Is there a way to import selected tables from a database, but not the entire database?

My site is still very much broken in several of its features, (search, payments, instant messaging) and not really fixable by me after the dozens of apps and updates we have gone through over the years. Wondering if we could install a fresh, clean copy of the latest software, and import only the usernames, passwords and profiles?

If all you want is the profiles from the old then yes, that is easy.  There are two ways you can do this.  What you want to do first is to export the Profiles table from the current site.  You also need to take note of any fields you have added to Profiles that are not part of the standard installation.  Install the new site and then add any fields that you added to the old site; you want all the columns that are in the old Profiles table to be in the new Profiles table.  Then import the old table.  I am going to skip the second way because what I outlined above ensures that other information stored on the fields is there; such as the type of the field, checkboxes versus select, The other way may cause some issues; maybe not but better safe than sorry.

I should have first asked which version you are coming from?

We have the latest version of the software.

It's just that we've used boonex since '09 and so much has changed that even trying to uninstall one of the problem apps breaks the whole site so we can't even reinstall them to fix it.