Yes, the supplier's recommended depth means that there is enough soil on top of the bulb to protect it during the winter and keep it from emerging too early. You should follow the recommendation. This means that if you have a bulb that is 5 centimeters tall, you should have 10-15 cm of soil on top of it.
That said, I have planted some very hardy bulbs shallower than recommended with great results. We have severe winters where I live, but daffodils seem to do especially well here. I've planted some bulbs just 5 cm under the soil, with the understanding that they may not naturalize as well, but if they survive, I'll have flowers at least a week earlier than I would have otherwise. The bulbs I plant shallowly like this are always the first ones to come up in my neighborhood by far and are very welcome after the long winter. It's worth the risk of losing them over the winter. I'd only do this with very hearty flowers, though.
As for your bulbs that are sprouting, there is no harm in planting them, but they may not flower in the spring.