Q.Amaryllis
After the blooms die, should I pick them or leave them on the stalk?

You can trim away the faded flower.
Here are some articles that will help you with care of the bulbs after blooming.
https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-care-after-flowering.htm
https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-care-instructions-how-to-care-for-an-amaryllis.htm
https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-bulb-storage.htm