Let's go right back to the beginning and ask why you want to grow apple seeds? Is it just a fun experiment or do you expect to get apples? If the latter then you need to understand that you will not get apples like the ones you found the seeds in. To get a specific apple variety you need to buy a named plant. Seedling apples may or may not produce palatable fruit. But they will not produce fruit like the parents.

Now on to the next point. If you live somewhere with cold winters by far the easiest way to grow apples from seed is to sow them in pots and leave them outside over the winter. Protect them from animals with wire netting or something. Forget them until the spring and then check for germination.

No cold winter? Now, and only now, do you need to worry about stratifying them in the fridge. And the key here is damp, not wet, cold. Sow the seeds in pots and place the pots in the fridge. Or place the seeds in a baggie of damp peat/mix and place in the fridge. Previous threads about growing apples from seed.

Rinko - a few years ago after he howled with laughter when I winter sowed 500+ milk jugs of perennials, my neighbor brought me 4 apple tree seeds and 10 pear tree seeds from his orchard. Around the 3rd week of March, I handed him 4 apple tree sprouts and 10 pear tree sprouts. The trees are now taller than he is.

Check out the Winter Sowing forum here on GardenWeb. It works. It's (mostly) organic. It's low-impact environmentally speaking and it's something fun to do in the cold winter months.

BTW - no fungus worries/issues.