I was at the range with my new-ish FNS-9C shooting Federal 124gr FMJ. I probably shot 50 rounds with the FMJ with no issue. I decided to run some Hydra Shok through it and all hell broke lose. 4/6 times the slide failed to lock back, and 2 failures to feed. I tried every magazine I had and the slide lock failure occurred with each one. The first failure to feed was simply the action not picking up the next round, thus resulting in a "click." The second FTF was more drastic (picture shown below).

Note: I decided to get some Hydra Shok because my LGS does not have Federal HST in stock. I have only fired 5 rounds of the HST through my FNS-9C... with no issue. I am aware that I should fire at least a box of the HST if I plan to carry it, but I am waiting for it to be in stock.

My question is: What would have caused these malfunctions? I don't believe it's the magazine, and I'm scared it might be the firearm itself. Are Hydra Shoks known for malfunctions? Any suggestions?

Break in the gun somemore, But polish the feed ramp to a mirror finish with a metal polish and a buffer pad on a dremel.
The feed ramp needs polishing if you feel the magazines are good.

I shoot at least 300 rounds to break in a new gun.  I polish the feed ramp too.

Guns are like people.  Sometime they meet ammo they just don't like.  Move on try another brand.

I have an FNX-9 and mine doesn't like Winchester hollow point rounds. It shoots Wichester FMJ and NATO rounds fine. So I use Speer Gold Dots, Remington Golden Saber, or Federal HST instead.

This is especially true with compacts, smaller guns have less tolerance of different ammo.  Find an SD round that runs well in your gun, stock up on it.

Good idea.  As a side note, until it is broken in, and until you get your chosen JHP (or an alternate) to function properly in the pistol, if you need it for defense, 124 FMJs that feed fire and extract are far superior to any JHPs that won't.  

I am at a point in my hoarding....er, collecting, that I never have to worry about needing a new gun for defensive purposes, but way back in time when I started out, before I was able to lawfully carry about, I was able to keep one in my car, and bought a Ruger P-90 for that task.  For the first six months or so I owned that thing, I wasn't able to scrape together the money to properly vet it with good JHPs (well, not really good--I was young and dumb and thought the lightest bullets at the highest velocity was best for "energy dump" nonsense, but I digress,) and so I left it loaded with 230 FMJ that I knew worked.  (And which were incidentally probably a better choice than the 185 Cor-Bons I later loaded it with.)

Short stroking.  Slide hits case ahead of rim and sticks because slide never goes back far enough.

Not the ammo.  

Gun too stiffly sprung, not lubed, new too new, or whatever.

Not locking slide is key symptom.

The FedHS 124/147 series is as good and standard an ammo as exists. Gun is fubar.  Anything that works, works with FedHS.

It runs Glocks, Smiths, Ruger, and an FN HP with out ever ever a malfunction.

Guess:  beautiful photo shows a bone dry gun.  No hint of lube in a perfect picture.

And not the slightest wear on locking surface of barrel and slide.

So... what do you recommend? Just oil/lube more? The weapon was pretty dirty when I ran the Hydra Shoks through it, but I did a proper clean and lube session afterwards. I will admit that the slide is a lot more stiff than any other firearm I own/racked. I have read that FN pistols are incredibly tight and you can leave the slide locked back for a while to loosen it up. I'm hoping it will loosen up over time and doing so, along with putting a couple hundred more rounds of ammo through it will fix these issues.

I know nothing of current FN pistols having never shot one.

I shoot clean well lubed pistols and they work.  Then I clean them and lube them before I shoot them again.  

Not bleached bone dry.

Your photo has nothing to do with the feed cycle.  The slide did not travel far enough to the rear to allow the top round in the magazine to rise ahead of the slide push corner.  So the slide just sticks in or ahead of the extraction groove.  Screwing with magazines  or interior polishing is not the answer.  Modern guns feed so directly into the chamber from the magazne that polishing usually helps nothing.

Strip the pistol. Look for clattering or other odd metal marks.  If none, lube is the probable answer.  See your manual on how.  

(  Lubing the top of the barrel hood and slide locking surfaces is standard.  Your photo stands out as dry. )

Locking slides back is a common suggestion, but I have no clue if it works.

I usually hand cycle the slide all the way a couple hundred times and do so another fifty to dry fire a new gun.  The rough edges are gone before wasting ammo.

If it doesn't work after that, there is some issue existing.

600 rounds of .380 in a BodyGuard is the last time I waste ammo breaking in a lemon just to be told the slide rails were wearing out.

Everyone has already said it, tear it down, polish the feed ramps, clean the hell out of it, lube it lightly with quality gun lube and try it again. If it still wont feed and fire reliably, move on to a different ammunition.

It is an FN that you paid decent money for.   If it does not feed factory SAAMI spec ammo like Hydashock, I'm sending it back or bitching at FN.  Not your job to do gun plumbing.   That's just me.

This post is on the money.  The gun does look dry and also very new.  Lube properly, but if it does not run HS I'd send it back.