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Non-stop begonias are tuberous begonias so they will form tubers. It may take 3-4 months to reach flowering stage.

They are very similar to propagate to the angelwing begonia. You can take tip cuttings, dip them into a mixture of a drop of honey,and a crushed aspirin tab in warm water just enough to turn into a thick syrup. This will be the rooting hormone. Stick this in moist seedling or potting soil, and spray when it starts to dry. They will root within a week or so.

Here is an article to rooting many plants:https://www.gardeningknowhow.com/garden-how-to/projects/rooting-plant-cuttings.htm

Unfortunately there is not really a way to tell, until you have growth.

You can start them in pots and label as they grow.

https://www.gardeningknowhow.com/ornamental/flowers/begonia/growing-begonias-from-corms.htm

I always use something small such as a chopstick, or other similar instrument. Wet the soil, and gently wiggle it out with the stick. Make a hole in the new soil that is slightly wet, and simply place it in, pushing the soil around the stem. After, you can gently water them to set the roots in.

Here is an article on their care passed this point: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm

It sounds like your plant has become leggy and could use a shape up!

Use a small scissors or garden snip and trim the plant back. Cut the stems down to a leaf node.
You can trim the plant back to about 3 to 4 inches from the soil.
Move the plant into a brighter light area after pruning.

https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm

That is a begonia called "Little Darling". It looks like the soil is very wet. This is likely the cause. Make sure that your container has plenty of drainage so that excess water can run out. You will only want to water the soil when it is dry at least down to the first two inches.

This article will help you: https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonia-leaves.htm