How can I prevent interior French Drain from further eroding soil under garage slab?
Using perf pipe allows water out just as it allows water in. The water stain half way along the garage wall could be from water backing up and out of the perf pipe. I like the solid pipe idea for "transporting" water, if you're not trying to "gather" excess water. Here we can buy one-half perf and one-half solid pipe. We install it with the solid side down and when the water table gets high enough to reach the mid-point of the pipe, it flows into the pipe. We just install the mid-point of the pipe at the level we want to keep the water table at. 

How about moving the sump away from the garage to the other end of the house? Then, when it starts pumping, it won't erode (suck) all the soil from the surrounding area out the drain. Also, maybe a sump pump that moves water slower or comes on more frequently so as not to impact the drainage system so severely each time. (It could cut down on the washing of the "fines" (soil or sand) out of the french drain and from under garage slab.) 

If you don't have a perf pipe along side the garage, you'll run the risk of water seeping into the garage. 

Also, look at separating your water. Remove all your downspouts from this collector system and use it strictly for subterranean de-watering. Rain tends to come quickly and thus flood the system...perhaps backing up and out of the perf pipe. 

Erosion is caused by water "washing" across a surface. If the material under your garage slab is dirt (or sand), then it'd be easy to erode. We use crushed rock. Can crushed rock be placed back under the slab, or is that too difficult?

Last thought: hopefully your installers used washed round rock (or crushed rock WITHOUT the fines) for your french drain. The key here is without the fines. For material to wash away from under the garage slab, it has to have a place to go. If it's confined then it won't wash away. However, if the french drain was installed using fines, then the fines in the french drain would wash into the perf pipe allowing the material under the slab to "sluff" away. Dumb!!! Are these guys licensed to do this kind of work? At least you have filter fabric encapsulating the entire french drain, right? 

I'm not sure what the solid sheeting between the garage and house is suppose to do.