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The offshoots are additional corms, which can be dug up and stored with the rest of the plant corms for replanting next season. While you can also save the seed pods and plant the seeds as well, the plants will take longer to mature so most people simply cut them off and discard.

Harvest flower spikes just before the top blossoms open, usually within three to four months of planting. The top blossoms will open indoors. Cut them in the morning or evening hours, angling the cut and immediately immerse them in water. Here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/flowers/fgen/how-to-keep-flowers-fresh-after-cutting.htm

I have always found fall is a good time to thin and transplant glads......and yes, trim your glads. before replanting them.

I would hold off on cutting the foliage back. While I know this can be unsightly, the plant depends on its foliage to make and store enough energy for bulb production and blooming next year.

If you are planning on storing the corms, this article will help: https://www.gardeningknowhow.com/ornamental/bulbs/gladiola/storing-gladiola-corms.htm

Yes, it is fine to soak the corms in this type of solution. It makes for healthier plants the next year. A solution made with Lysol is most commonly used, but any disinfectant can be used. A very weak bleach solution would work as well.

Plant in the spring once the threat of cold has ceased and the ground has significantly warmed up. Glads require well-drained soil to prevent them from rotting. To improve the quality of your soil, you can work in some sand and compost. Depending on the variety, most are planted in a sunny location of the garden, usually half as deep as their expected overall height (or 4-6 in.). Always place bulbs with the points facing upward, while corms should be placed with the depression facing upward. Cover with a layer of mulch and water thoroughly after planting.