Hi exfx and welcome to slackware.
I'm probably not the best advisor in your case, since I belong to the minority(?) of slackware users who rely on gslapt/slaptget for upgrading.
slackpkg, with the extension slackpkg+, should do the job. But since you want to follow upgrades in 3rd party repos and have gone multilib I advise you to carefully read AlienBob's exhaustive slackpkg+ documentation here:
https://alien.slackbook.org/blog/int...-repositories/
Also, read his multilib documentation:
https://wiki.alienbase.nl/doku.php?i...kware:multilib
Remember that new packages (dependencies) are sometimes added to the slackware -current repo (or, of course, to 3rd party repos). You'll catch them by running "slackpkg install-new"
When on -current it's always a good idea to check the changelog. Look out for added packages and upgrades that affect other packages (e.g.the recent python3-3.8 upgrade)- those may affect 3rd party programs that will need rebuilding. Also look out for the line "Shared library .so-version bump." which may affect 3rd party packages - especially icu4c and boost. AlienBob's icu4c-compat and boost-compat packages are indispensable safeguards against package breakdowns.
It might also be a good idea checking this forum before upgrading, seeing if other users have run into problems with a recent upgrade.
If an upgrade creates problems, downgrading might be an option. Older packages can be found here:
http://slackware.uk/cumulative/ but use this option with care or you might be in a deeper hole than before
As to your second question: if you've downloaded or built a .tgz or .txz package and it works OK, it should be fine transferring it to another computer with identical setup and install it there, using "installpkg [packagename].txz
If you manually download a .tgz or.txz package from a trusted and reliable source - the official slackware repo, AlienBob, Ponce - you can be sure that it installs correctly and in its proper places.