I've been aging in the food fridge since I started making cheese last year as well. Its not ideal, but you work with what you have until you can improve your set-up. I've commandeered 1-1/2 shelves so far, and the natives are getting testy. I am hoping to add a dedicated wine fridge in the spring, but I'll say I'm satisfied with my results so far. Certainly better by a long shot than no home crafted cheeses at all. I whole-heatedly encourage you to keep at it.
The harder cheeses will take longer to age and I suspect the final results will always be a bit different than with traditional affinage. Neverthelessless, the slower aging has worked to my advantage in having finished cheeses in March and April that would otherwise been eaten sooner. Our main milk supply is seasonal with our goats. I like to think that I'm less likely to miss a cheese's peak.
I've found that bloomy rinded and washed rind cheeses tend to do well in the cooler temps. The washed rinds take a bit longer to get going though.
The biggest challenge I have is cross contamination of blue molds and geotrocum strains. There are lots of crazy things growing on the fruits and veggies that revolve through the fridge. The geos aren't so bad, but the wild blues usually have less than desirable flavors. Some sort of ripening box is a must if the cheeses aren't waxed or sealed. I use ziplock bags with some success when I'm short on boxes. Vinegar and salt scrubs and a deft hand with a toothpick help keep unwanted molds in check. On a a firm, smooth surfaced cheese I sometimes let the wild things run, knowing I can remove the rind if I find it too strong. I've got two larded cheeses aging now, and that seems to thwart wild molds well. Molds that do grow on them come off easily and then fresh fats can be applied. I use 3-5% brine to dab off blue from bloomy rinds, but that is usually a losing proposition since the rind surface is so fragile. Best to do what you can to keep them isolated from cross contamination from the get go.
You can use coolers and ice packs to simulate cave conditions too, but I haven't tried fussing with that yet. Seems it will add a lot of daily maintenance that I can't take on presently.
Maybe Santa will bring me a vacuum sealer for Christmas. I'm afraid I've been too naughty for the wine fridge this year. One can hope, though.