It's your lucky day. You could call me a stone expert! My business is stone masonry. I'm third generation, and have been running around at the job sites messing with rocks a long as I can remember. I went full time as a mason in 1994. So... I'm your rock man.
On rocks and leaching/seeping: I'll try my best here not to offend Kryten, but he's kinda hit and miss through out his reply. He did say he's not up on rock chemistry, so that lets him off. No big deal. Ok. When rocks leach, it's minerals and metals that are released. There can be chemical issues, but those would be from surface contamination instead of seepage.
99% of all rock that leach are sedimentry. This type of stone forms from layers of sediment at the bottom of the ocean. The reason this rock leaches is because the minerals and other elements suspended in sea water get trapped in the sediment on the ocean floor. The imense weight of the ocean compacts the sediment into solid stone. We're talking millions years ago here. As the tectonic plates move, oceans dry up and expose the sedimentary rock. This type of rock leaches so easily is because it is still the exact same thing it was millions of years ago. There was no actual phisiological change, only compression. Limestone is sedimentary, and so is sandstone. Sandstone does not leach however, as it forms under different geological conditions than limestone.
You also have igneous rock, which is produced by volcanic activity. This rock started out as something else, but the extreme heat has melted it. This heating and cooling does produce a physical change in the rock. Lava rock and granite are igneous. I have never known ingeous rock to leach or seep anything.
Then you have metamorphic rock. This type of rock forms exactly like sedimentary rock, but only where massive amounts of organics were trapped. These organics act as a catalyst for physical changes in this rock as well. Again, I have never known metamorphic to leach or seep anything. Coal and slate are exaples here. Slate is actualy one of the most leach-proof rocks there are. Think about it, they use it for roofing...
Now on your nitrate...
Yes, a lot of people use rock to create caves and other natural looking aquascapes. The difference between you and them bumblebee is surfave area. Nitrifying bacteria grow on every surface inside the tank. Someone else might have even more rock than you have weight wise, but less than half of the total surface area your little rock wall has. Basicly, you discovered the problem yourself.
Solutions...
What type of filtration do you have? Your filter media could be a nitrate factory. The tank's about 4 months old? Have you cleaned the filter out? If not that would help too.(remember never clean all your filter media at once, and when you do clean it, scrub it clean in the water cange water you take out of the tank.) I think you could put the rock wall back up after the tank re-stabilizes. I wouldn't use silicone filler, not at first. Go without for a while and try this...just simply using the siphon hose along the cracks in the wall to suck the waste out of the cracks. It should work quite well, unless your slate is cut and stacked with near zero tolerance. If the crack siphoning doesn't work, silicone filler is an option. I'm afraid though that if the silicone filling is not done thoroughly enough, it could actually add to the problem.
How often have you been water changing? I see you did a few after noticing the problem, but what about your normal water change schedule? You might consider altering your schedule by changing water more frequent in hopes of preventing another build up.
Some plants do consume nitrate, (some algaes do too) but they also consume a bunch of other stuff too. Plants can be hard to take care of, but if you had a CO2 system and power compact lighting you'd have 'em growing like weeds.
Nitrate reduction is a tricky matter. Ya just gotta find what works best for you.