The sunnier it is the stronger everything, even flower stems, will be. Feed that bulb, pizza, chocolate, anything it will eat, and you'll get more flowers.

"I do have one random question, however: is there any way to encourage a bulb to produce more flowers per bloom cycle?"
Yes, the bulbs develop flowering buds in the cold period. Those can originate from the axillary bud of each fourth leaf. So in order to get more blooms you (of course! ... but in this case it's really closely linked) have to have more leaves! ;)
So give the bulb full sun, good draining medium (so you can reduce the risk of fungus) and water periodically - but then give the bulb a good gulp with some chemical fertilizer if you have them in pots!

"Also, what controls the length of the flower stem?"
I think there are diffent factors controling the lenght. Mostly it's the genes but you can influence it via temperatures. There have been some examples here on the forums where people showed pics of their blooms directly atop the bulb's neck. Those where influenced by colder temperatures if I recall correctly.

I understand your excitement! In my early amaryllis fever - I bought many bulbs, which after blooming shrunk to almost nothing in some cases. I did not know about the after-care required. And certainly no reblooming was seen. But, in subsequent years, mostly the last 3, I have been taking the newer ones outside for summer in the sun and feeding them as well. Lo and behold - I have reblooms - not all, but much better performance than before. Even some small bulbs have surprisingly produced very large scapes. Last summer, my Red Lion bloomed in July! And forget about Christmas reblooming - IMHO - it ain't gonna happen - just be happy when they do decide to bloom. And I do love such surprises! If you want holiday blooms - buy new bulbs (yes!)

Thanks for the advice! I know I read somewhere on this forum that there's a general ratio between number of leaves and number of flowerstalks (3 or 4 to 1, I think), so I'll keep that in mind.