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Blooming times vary depending on location, temps, soil, etc. Generally, following their spring planting, they should begin blooming in summer. Most will bloom anywhere from 3-6 weeks with adequate care, though by October yours may be nearly finished for the season (not at their peak). For additional information, these articles should help: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/growing-calla-lilies-and-care-of-calla-lilies.htm, https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/make-calla-lily-bloom.htm

This is fine as long as it's not an everyday thing. Once they're in the ground, you should keep the soil moist but not too wet. If they remain too wet, they will rot.

Yes, if you live in a zone lower than 7, you should really dig them up and store them over winter. This article will help: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/calla-lily-winter-care.htm

It is generally better to wait until the plant has finished blooming so that it will cause less shock while it re-establishes itself in its new location.

Unfortunately, the mushy bulbs have died and should be thrown away. It may be too late for these, but this article can help you in the future: https://www.gardeningknowhow.com/ornamental/bulbs/bgen/preparing-bulbs-for-winter-how-to-store-bulbs-for-winter.htm

This article can help with that: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/green-calla-lily-blooms.htm