To address your specifics, yes bulbs can be stored in their bag in refrigerator. However, if not in their forcing pots, the time spent therein doesn't count in the actual cold period requirements to force them. Except for hyacinths, hyacinths can be left in their paper bag in the refrigerator with the time used before potting or placing over water counting as part of their required cold period.
I always place tulips and daffodils into pots, water them, let them sit for a week (to get the roots started), then into the dark, cold period in the refrigerator.
Don't forget to put no fruit in that refrigerator due to ethylene gas produced which is harmful (terminal) to your bulbs.
Apologies for a not sooner response.