The comment from David Carlisle is the answer: older LyX versions cannot read files that were saved from newer LyX versions.
This is not a Windows vs. Linux issue: You would have run into the same problem if you used a newer version of LyX on Linux to create files, and then tried to open those files on Linux with an older version of LyX.
This is an introduction to an issue that you will face a lot of Linux (at least, on Mint). Many versions of software in the repositories are years older than the newest version. To get the newest version of LyX on Mint, you should consider using a PPA:
From what I understand, Mint users can use Ubuntu PPAs. I suggest using the stable PPA. Instructions are here.
Longer explanation: LyX version a.b.c can for the most part only open files that were saved with LyX version x.y.z if a>=x and b>=y. For example, LyX 2.1.2 can open files created by LyX 2.1.3, but not files that were created by LyX 2.2.0.
An exception to the above is the last release of a series can open files from the next major LyX release. So LyX 2.1.5 can open files created by LyX 2.2.z, for any z, because 2.1.5 was the last release in the series.
Why does this restriction hold? Well, when LyX gets a new feature, it might need to change its file format if the older version can't support that feature.
By the way, welcome to Linux! It is normal that it takes you some time to figure out issues like these, but after you figure out a few of the transition issues you might find that you see your productivity (slowly) increase (some disagree though). Note that there are many Linux + LyX users on the LyX mailing list, including some knowledgeable Mint users. Feel free to subscribe.