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I find Calla Lily does not do well inside. Mine do very well in the garden with a sunny wall behind and a reasonable supply of water.

This article offers a few insights: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/drooping-calla-lilies.htm

There are many things that could be eating your calla lily leaves. If you've eliminated slugs, that still leaves a host of caterpillars, grubs, beetles, and grasshoppers. You can do some detective work at night with a flashlight, or dig around in the soil to see if anything is curled up there. Even if you can't identify the culprit(s), you might try spraying the plants with neem oil. https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

This article should answer your question: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/green-calla-lily-blooms.htm

The pod is actually the flowers, hundreds of very tiny flowers. The white part is a kind of bract, which are modified leaves. While under very special conditions the flowers might become fertilized and produce seed, this is not likely to happen under home conditions. So no, you can't plant the "pod' and grow new plants. You can grow new plants by separating and planting the rhizomes, what some people think are bulbs.
Here is some information on drooping callas: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/drooping-calla-lilies.htm

There are dozens of varieties of calla lillies, and most of them will bloom for 2 or 3 months, the flowers lasting a couple of weeks. Of course you can cut the the flowers -- cut the stem as close as possible to the soil, at the point where it emerges from the leaves. After the flowers have bloomed, you can save the rhizomes, and plant them again next year for more flowers. This article will tell you how: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/growing-calla-lilies-and-care-of-calla-lilies.htm