No matter what I do, game in full screen always shows two black bars, or big black frame.
What should I do?

Uncheck the 'Keep aspect ratio' in the Graphic mode setup (advanced tab).
Though it is possible this function is bugged (my own experience) in which case you have to open the dosbox_swars.conf file with notepad and search for the line 'keep aspect ratio=' and make it false.

edit:

Though keep in mind that with the black borders you play with the original aspect (4:3), removing the black borders will just stretch the whole image and mousepointer movement might feel a bit awkward.

I couldn't find the line 'keep aspect ratio=' but I suppose you mean 'aspect='. Setting it to true or false doesn't change anything.
Does the scaler have anything to do with this?

No scaler only affects filtering and windowsize AFAIK.
Do you have your native displayresolution selected in the Graphic mode setup?

Yes.I've tried changing the "fullresolution=" line to both "desktop" and my native resolution,but nothing worked.

I'm starting to run out of ideas, selecting a different display output might help.

Yeah, me too. Both Opengl modes crash and I've tried every other output. At this point I'm starting to believe it might be something wrong with my PC.
Thank you so much for your help, though. If I ever find a workaround, I'll post it here.

No problem, one thing I still like to ask though, is it just Syndicate Wars or are other Dosbox games affected too?

I can't remember a single DOSbox downloaded from GOG doing this. Anyway, to be certain I wasn't mistaken, I downloaded Tyrian 2000 and fired it up, only to see that the screen is perfectly stretched and with no black bars.
UPDATE:
Since Tyrian 2000 was perfectly working, I copied the settings of "dosboxT2K" directly into the Syndicate config txt, "dosboxSWARS", after making a backup copy of it obviously. The result is in the attached picture.

TL;DR: If you have a problem setting up a certain DOSbox game, try recycling the CONF file of another (working) DOSbox game.

For now, this should work. In the very next days I'll try to understand which line or which set of lines was creating the problem. Thanks, Strijkbout, for forcing me to think this through.