There are so many orchids that fit that description. Lets try to narrow it down bit: 1. Are the flower stems completely different structure from the leaves? or do the stems first come with leaves and then those same stems produce the flowers after the leaves fall off? 2. What do the leaves look like? and how are they arranged? Do they grow all from a central "crown" or do they grow along a cane stem? 3. Do you see anything that look like "bulbs" at the base of the plant? A search for "rat tail orchid" got several hits for nobile dendrobium and bulbophyllum. That at least narrows down our search to uhm... a few hundred varieties? :) Try to post a picture of the orchid, that will make it easier to ID. You mentioned the old flower stems produce new baby orchids (keiki). Dendrobiums can do that, I'm not sure about bulbophyllums.

Your plant is a Brassovola. Mine grows in full, Florida sun inside a screen patio (so I suppose the screen gives a bit of filter). In NY I would have it in full SW window and would get a few blooms. Checked it yesterday and it now has 14 spikes. Loves the Florida sun/heat. I am growing it bare rooted. Unfortunately, something ate 5 spikes. Gradually, move it into stronger light. You will get flowers as your plant looks healthy. This is not a good shot but gives you an idea about light. Jane

It is a Phalaenopsis. Culture notes are in the link. AOS Phalenopsis notes

I never saw hairs on a Amaryllis bulb. I grow them in the ground in Florida. I have no idea what that is, but I would repot it in a very well draining mix. Whatever it is it stores water. The size of the bulb and the thickness of the leaves. My guess this plant likes dry conditions and high light. I would repot in a cactus type soil and move it gradually to a very bright, sunny window. Grow it dry, only water when it is completely dry. I'm guessing, but that's what I would do if it were my plant. Good luck and let us know what happens. Jane