Something I have never understood about Linux is that if you're on lets say Arch, and you need to install a .deb package because there isn't a package for Arch, I don't know what you do. I'm not sure if its possible for you to install it, really complicated, not recommended, or not possible. Linux is Linux, x86 is x86. You should be able to run these packages interoperably, even if you have to install a few dependencies you wouldn't normally have to. Is this right?
@kira Yes, you're right. As long as the software was compiled for the correct system architecture and you have the appropriate shared libraries on your system the software should run regardless of the distribution.
Now, regarding installing this software, in this kind of cases I would rather compile the software from the source code if that is available as that makes it easier to update as long as they provide a half decent make file.
However, compiling software may be tedious especially if it's something you never did; thus you may prefer installing the software from the package provided on another distribution.
The reasons why you have different software packaging methods and repositories are complex, but mainly boil down to different philosophies an requirements on update schedule and stability.
This makes it so that different package management software are not compatible with each other: you can not install dpkg packages with pacman. However, what they do is just place a bunch of documents in an appropriate directory, that can be done manually.
If you look at the structure of a package file, it's generally just an archive which contains all the binaries, documentation and libraries associated to the software and some additional information regarding where to place this stuff on the system and what other packages are required to make it work properly.
Thus, you can unpack the archive and put the files in their place https://www.cyberciti.biz/faq/how-to-extract-a-deb-file-without-opening-it-on-debian-or-ubuntu-linux/. If you decide to go this way, I would advise using a utility such as stow which will allow you to keep files in your system organized together and allow you to easily uninstall the software.
Now you can test if the software works, if it does not work it's because you're missing some library, thus use your package manager to install those as well. If it still does not work, then the libraries provided by your distribution are not compatible with that software; in which case you would have to manually install the appropriate library version as well.
