And if so, do you have a previous martial-arts background?

Yes.  But what is it you really want to know?

1.  There are different versions of Systema in Russia.  Vlads version is sort of a far left wing version.
2.  There are several 'spin-offs' from Systema here in North America.  Kevin Secours (Canada) version is very good.  I strongly recommend his videos.  He is able to organize and explain very well.
3.  Systema is usually taught in a random fashion, no sequence/progression.  That makes it more difficult to learn.
4.  Often a 'high level' version is taught but what is not taught is how one got to that level.  Again, makes it more difficult for the student.  It either 'clicks' right away or it doesn't.
5.  OTOH Can find classes where nothing is ever practiced past the very beginner level.
6.  The large amount of slow practice is very beneficial, however many classes never speed it up some and 'function test'.
7.  Training partners can easily become too co-operative without realizing it.
8.  There is much more work with multiple attackers and if the defender is on the ground being kicked, etc. than in other styles.  This is a big plus compared to other styles.
9.  The sensitivity and awareness work is very good and carries over to training in other styles as well.

I just started with it a couple months ago, and am trying to decide if I'm a "believer" or not.

Some of the drills are just starting to make sense.

The emphasis on "relaxation" and "tension" is not making sense. It's like the words are being used in an unconventional sense or something.

I like the random, un-sequenced progression, because with my bizarre schedule I can only make about half the classes.

Love the work with improvised weapons, multiple opponents, and unconventional positions.

Am very curious about Vlad. He apparently has a dark past that everyone hints about but no one talks about. What do you mean by "far left wing"?

Now that I've been involved for a few months, I can say this: a lot of the things on Youtube that looked fake to me are totally real.

My instructor demonstrated a "wave punch" on me.

My best friend told me it looked fake, like I dramatically over-reacted. He even phoned me later to ask if I was faking it.

It was all I could do to stay on my feet and finish the class, and then I felt ill for a couple days.