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Re-potting a mature plant should always be undertaken with care so as to avoid damaging the plant or the roots. The easiest way to re-pot your String of Pearls might be to just hold the plant over the new pot and remove the old disintegrating pot while letting the soil fall into the new pot. You can find more information about String of Pearls plants here: https://www.gardeningknowhow.com/ornamental/cacti-succulents/string-of-pearls/string-of-pearls-plant.htm

If the "white" color is actually a powdery-like substance, it could be some mineral residue from the water you may have used to water the plants. More than likely the plants are experiencing a little "shock" from the replant and once they adjust to the new pots and conditions they should be fine. Be sure to not overwater and hopefully you placed them in a sandy-type soil.

This article may help you:

https://www.gardeningknowhow.com/ornamental/cacti-succulents/string-of-pearls/string-of-pearls-plant.htm

Shriveling or drooping of succulents is common and usually caused by watering problems, usually overwatering. Excessive water can damage the roots, causing the plant to be unable to absorb water and leading to wilting. String of pearls would only dry up from underwatering if it was left alone without water for weeks or months. Proper watering procedures are described in this article:

https://www.gardeningknowhow.com/ornamental/cacti-succulents/string-of-pearls/string-of-pearls-plant.htm

First, if some of the pearls near the base are still healthy, cut off the dried up pearls to leave only the healthy ones. This will give the roots less foliage to support. Then, work on correcting the watering schedule. Also check this article:

https://www.gardeningknowhow.com/plant-problems/disease/treating-root-rot-gardening-tips-for-housplants.htm

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There is very little detailed information on sowing the seeds from String Of Pearl.

I did find that you should sow the seeds directly on top of your soilless mix. Keep moist and warm. Germination may be very slow.
Do not allow the seedlings to dry out.
Propagating from cuttings is the recommended method.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/string-of-pearls/string-of-pearls-plant.htm

Here is an article on the care of 'String of Pearls' that will help you be sure you are giving it the right care: https://www.gardeningknowhow.com/ornamental/cacti-succulents/string-of-pearls/string-of-pearls-plant.htm

During the life of the plant you will come across some dead strands every now and then. I would recommend trimming off any dead stems and pearls, as well as any stems that have lost a lot of their ‘beads.’

Yes, this succulent plant usually does well indoors. The main thing to know is that string of pearls prefers dry, desertlike conditions, so it doesn't need to be watered very often, and it prefers the type of potting soil used to grow cacti:
https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/cactus-potting-soil.htm

This article will tell you more about string of pearls plant care:
https://www.gardeningknowhow.com/ornamental/cacti-succulents/string-of-pearls/string-of-pearls-plant.htm