Well.. I'm not familiar with that particular pedal unit, but I do use some Behringer gear.  I'd say I rate Behringer as "ok to good" for the price, so far as I have seen. 

RG316 is an rf cable, so if anything, it should be better than you need for audio shielded cable.  I've never worked with it, so I don't know how stiff it is or how well it holds up to repeated bending.  But it certainly looks worth a try. 

With playing through a stereo, your signal is mono.  You could use a splitter, but it won't actually be stereo, since the two channels will be identical.  If that isn't a problem, then yeah, it will probably work.  Whether you'll like the sound of that or not is another matter.  Same goes for computer speakers.  In both cases, you'd need to make or get an adaptor to go between the box and the stereo or computer speakers, obviously.

You've kind of hit the point of "have to try it and see" to know for sure.

Ok, let's start with possible downsides.  I have never played this amp, or any of it's bigger relatives.  So I can"t tell you for sure how it sounds.  But the reviews are pretty good and most Behringer gear I've owned I would rate "ok to good".  Which would make it definitely worth the usual amazon price. 

I also didn't see any mention of anyone using it with a piezo pickup, direct.  I didn't read all 40 some reviews, maybe you did and somebody did mention it.. But unless they did, it probably isn't actually made for taking a raw piezo signal.  If it doesn't actually say it has an input for piezo pickup, then it probably doesn't.

Yeah, at the worst, you'd end up using the 15$ ebay wonder to get the signal up to the usual impedance and level of a magnetic pickup, so that isn't such a huge deal.

The features on it are ok, but kind of minimal.  The main selling point is that it at least says it is better for acoustic guitar. 

When you start thinking about buying an amp, you really should get to a music shop so you can actually hear how they sound.  That's not a couple bucks or 15$ or even 40$.  I'm kind of iffy on my thoughts about a practice amp for an amplified acoustic violin.  To be honest, an acoustic violin is loud enough that you don't need one to practice, and if you needed enough louder than they naturally are to be able to play a gig or jam with a band with loud electric guitars or something, a practice amp just isn't going to cut it.

If recording is what you are after, you can get a nice but inexpensive studio mic, even one that will connect directly into your USB jack for computer recording for around the same price as you are looking at with that amp.  Even just a condenser mic made to go into computer that costs maybe 10$ or less can be mounted using the same idea you had for strain relief and you can run that direct into your computer.

Before I go saying it sounds like you might be losing sight of the forest because of the trees, I'll ask a question.  What do you see yourself doing with the rig you are looking at putting together?   It sort of feels like you have some playing situations in mind or some ideas on what you might want the rig to be able to do.  Clarifying that could help on what advice to give.

Oh, and the tube-vs-transistor controversy?   I'm going to be honest with you.  While I personally like tube amps a bit better, the biggest differences are in how they distort and how they act at high volume levels.  When playing them clean, there isn't that much of a difference to the average listener's ear.  And there are some very nice sounding transistor amps and some very crappy tube/valve amps.  Pros use either and or both.  It largely comes down to what *you* like.

An electric violin of the "solid body" sort is very quiet.  They only get loud when amplified, and then one can always turn an amp down to a suitable level.  They allow for using assorted interesting effects like an electric guitar uses.  They also can plug directly into a computer or other recording device when you want to record, and since it is a direct connection, you don't get the sound of the computer fan or the neighbor's dog howling or whatever.  I would say their biggest problem is that they don't really sound like an acoustic violin.  They can sound good, but they just don't sound the same.

Acoustic violins are a lot of sound for such a small package.  Mine is a bit louder than is convenient for night-time playing.  I can play it quietly if I use pretty much no bow pressure, but enough pressure to actually get a good tone and she is a loud little thing.  I've never tried a mute because from what I've heard they affect the tone (as you mentioned), and I can just use my electric for playing when I need to be quiet.  But for portability, having enough volume to play for quite a few people all by itself, and definitely sounding like a violin/fiddle, nothing beats the acoustics, in my opinion.  One can't play around with effects on them like an electric, but what they do have is a very good range of usable musical sounds. 

Amplified/electrified acoustics, I am less sure of.  In some regards they could have the advantages of both, but also the disadvantages.  To my ears they don't sound quite like either an acoustic or an electric violin.  I haven't messed much with that sort of thing myself.  Since I have both an acoustic and an electric, it hasn't personally been of much interest to me as a concept.  The couple of crude pickup/mic schemes I posted are just a quick and cheap way to get a signal from an acoustic violin into a computer or recorder, if maybe someone has an acoustic and doesn't have a microphone.  Personally, I probably wouldn't put a piezo pickup under the bridge of my own acoustic, since I have no idea what it might do to the acoustic sound. 

But folks do, because it is what they want.

As to why does anyone need an electric?  Well, the best reason I know of is if they have sounds you like or they look like fun to play.  To my way of thinking, electric and acoustic are almost two separate instruments, for all that the basic techniques are the same.  For me, it is what I mostly want to play.   Having gotten an acoustic as well, I like it too.  But they are different and I kind of have to practice on both, same as I do with electric and acoustic guitar. 

I can't say for sure, but I would guess that electrified acoustic is probably actually a bit different than either one of them.