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Cut the flower stalk about an inch or so above the ground. Place them into water as quickly as possible after they are cut. This article will help you with keeping the fresh and lovely longer:
https://www.gardeningknowhow.com/ornamental/flowers/fgen/how-to-keep-flowers-fresh-after-cutting.htm

They create themselves. When you dig the bulbs in the Fall (if in cold area), you will see new bulbs. Some are very tiny but usually there is at least one that has some size to it. These are planted when you plant in Spring. If you are in a warmer area, they can just be left in the ground without digging. They can be divided in the Spring and planted where you want the new bulbs. Tiny ones should be planted pretty shallow. Please supply your location or planting zone when asking future questions since answer are different depending on that information.

Packing them in cardboard boxes (see article) and storing them in the garage should be okay, provided the area doesn't get too warm. Here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/bulbs/bgen/preparing-bulbs-for-winter-how-to-store-bulbs-for-winter.htm

The following article should be of some help to you: https://www.gardeningknowhow.com/ornamental/bulbs/gladiola/gladiola-disease-pests.htm

This is just a seedpod. You can cut if off unless you want to harvest the seeds; however, starting glads from seed will take much longer to grow than from the corm itself.

I would dig up one of the corms and see what it looks like. It could be either a fungus or a grub or rodent. The corms should be firm and whole. If they are, they should be usable, but I would treat them in a very mild bleach solution to kill any fungus that might be on them. If they are not firm, then they are dead and will need to be discarded.