You can remove old flowering stems but the foliage needs to remain in place. The foliage provides the ability to photosynthesize which the bulbs need to replace their energy stores to provide next season's flowers. It is recommended that it be allowed to ripen and dieback naturally - once fully dry and browned, it will generally pull or fall away from the underground bulb easily.

Tying, using rubberbands or otherwise bundling up the foliage is our human way of attempting to "beautify" or tidy up the natural process. It doesn't help and can hinder and slow the natural process and often promotes the development of disease and insect problems. It is best to just allow nature to take her course and if necessary, disguise the old foliage with the new growth of other plants.

Bulbs will easily grow up through mulch. A caution might be exercised with the specific type of mulch - anything that is too moisture retentive may promote fungal issues or rot emerging foliage.

Gardengal is absolutely correct, as always. I would add that the depth of your mulch is unlikely to affect the bulbs' abilities to make their way through it. I suppose it's possible to get it too deep, but you'd really have to go Way overboard. (Like maybe over 6 inches!)If you are a newbie, chances are you're not going to plant bulbs that are that delicate and sensitive yet. When you start branching out, this is a good issue to keep in mind, as Gardengal said.