Easiest way I can think of is to mount the Slackware DVD (or DVD iso) to a directory and then just run upgradepkg --reinstall --install-new /location/to/slackware-dvd/slackware64/*/*.txz. Once this is done, you should have all the stock packages reinstalled. Then, hopefully, it should just be a matter of using slackpkg to update the system to get the latest packages.

Keep in mind, since you showed multiple openssl packages (and there's the possibility other packages had multiple versions installed), you may end up with files that belong to no recognized package. This may or may not cause problems when compiling software that relies on those packages.

You may also be able to reinstall everything with slackpkg, but as I said before, I'm not too familiar with the program. You might be able to do it with one of the below commands (since /var/log/packages was cleared, it might require the "install" option instead of the "reinstall"):

Good luck!

If you notice the package location of each openssl package you will see that one is a patch and the other is the full package that had been patched. slackpkg will list all packages you query as well as their subsequent patches. You were mistaken not to read the path of the package in the output.

The following will show all packages and patches for the system.
The command you should be using with slackpkg to see what is installed / upgraded / uninstalled are:
If you cannot find the package by that name for whatever reason, then there is the 'file-search' option. This option will list the packages, related packages, and any other packages with files or dependencies related to that string query.
I suppose though the point is moot now. If you deleted all those files you mentioned and pkgtool is missing, you have bigger problems!

You should have checked your PATH variable as well as used the which command like so:

As I mentioned above shalt taste the slpkg? if you do not then fine.

I think I've figured out what went wrong. I cleaned out everything I could find with slackpkg logs, downloads, etc. Started `slackpkg update`, which did download a bunch of stuff. Then `slackpkg ugrade-all`, but at the end of that, I still had my old programs; e.g. openssl 1.0.1e, not 1.0.1m.

So, I tried hand-upgrading. That worked for openssl and for samba (4.1.0 -> 4.1.17), but after upgrading httpd from 2.4.6 to 2.4.12 I got "httpd: symbol lookup error: httpd: undefined symbol: apr_table_getm". So did `slackpkg install apr` which did upgrade that package, but httpd then gave an undenfied symbol error with mod_proxy. I couldn't find a way to upgrade httpd modules (the install of the new httpd didn't do it), so I restored everything from before I started messing, upgraded openssl and samba and left it at that.

Later, I installed Slackware 64 4.1 on a completely different computer from the same DVD. I did the `slackpkg update` and `slackpkg upgrade-all` (it ran for about 20 hours!) and everything upgraded just fine.

What I think happened was that after installing the 1st system and before doing the upgrades, I restored production directories from the original backup (this was an update of an existing system). Those directories included /var/log and /root which hold slackpkg info. I apparently clobbered enough config stuff that slackpkg was hopelessly lost. A lesson for next time!

What would be really nice would be a `slacpkg reset-all` that would put things back to the as-installed state, because restores happen! Or even a `slackpgk discover-all` which would examine all the packages and correct the package archives depending on the actual version installed.

If you overwrite /var/log/packages and /var/log/scripts with backups from other computers, then there is nothing you can do to fix that. These directories can not be re-populated from your computer's content.
It's no different when you kill a RPM database on a Redhat machine by the way.
Be more careful with the package database next time.