Grab the Windows version of GZDoom.
Copy your strife1.wad and voices.wad into the GZDoom folder.
Set the GZDoom.exe to 'Open with Wine', if it isn't already set to.

Now, whenever you wanna run Strife, simply launch the GZDoom.exe.
Also works with Heretic and Hexen.

As for the music:
Go into OPTIONS
Go into SOUND
Set it to OPL Synth Emulation.

I've done all this with Linux Mint 17.3 and Linux Lite 2.8. Works great.

I resorted to this because Strife Veterans Edition wasn't running on Linux Lite. So I tested my options with the original WAD via GZDoom for Windows.

Another ironic tidbit: the Linux versions of ZDoom and GZDoom require alotta jumping thru hoops to get them to work. But the Windows version of GZDoom runs just fine via Wine. Weird...

It may be best to contact support if you can't run Strife: VE, especially if your distribution is based on Ubuntu.

Using GZDoom via Wine seems like far more hoops to jump through than should ever be necessary...

Also, you could try asking on the GZDoom forums (no need to register) about your issues with native builds on Linux. One of the ZDoom developers works near-exclusively with Linux, so there shouldn't be that many hoops to jump through...

I simply skipped on the whole 'get online help' thing and went guerilla with it. I'm happy with the results.

LATE UPDATE:
The original Strife on Linux (Mint 17.3) via ZDoom for Linux.
Runs great.

Download the original Strife (included with GOG purchase).
Copy the strife.wad and voices.wad to your ZDoom folder.*
Launch ZDoom.
Select "Strife: Quest for the Sigil".
Enjoy.

ZDoom for Windows, Mac (including Power PC), and Linux can be found here: http://zdoom.org/Download

ZDoom folder can be accessed via Home/.config/zdoom

Expose .config by holding Ctrl+H I figured out how to get ZDoom for Linux to read my .wad files.

Home
.config
zdoom

Access .config (and other "." folders) by holding Ctrl+H

I know, duh, right?