As a prologue the writing is a bit difficult to judge. On the one hand it felt very formal and biographical, but on the other prologues tend to be granted more lenience. My biggest suggestion here would be to avoid writing as formally in subsequent chapters for voice and atmosphere purposes.

This is definitely a rare plot twist, and I'm intrigued to see where it goes. You covered enough info this chapter to make the reader ask, "Oh, shit, how's he going to manage this?" Looking forward to reading more!

There are a couple things I immediately thought when reading this, and the first was: "Are people under the age of 18 allowed to donate sperm?" After Googling that question, it looks like no. So you may want to address that if you decide to keep him 16 from the time he first donated, because I'm sure others will ask that question and it's going to affect the reader's suspension of disbelief. You can either up his age a couple years or go a little more into detail about how even though it was supposedly against the rules (I don't know if it's necessarily illegal), they decided to still do it for whatever reason (maybe he desperately needed the money and had no other option). The other thing that I thought of while reading this was that the basic premise sounds a lot like that new comedy with Vince Vaughn where it's a very similar situation: guy donates sperm and ends up with a bunch of kids, and a dude shows up at his house (dressed in a nice suit) to deliver that news. Just something to think about. This can obviously stray away from that plot and become its own story, but right now I'm a little iffy on just how similar it is to that movie.

Also, I think playing this entire prologue out in scene might be more engaging for a reader. Right now it feels a lot like plot summary, or back-story. If you delivered it through scenes or during the exposition and interactions between Jack and the other characters, I think it'll be a lot more attention grabbing and a lot better of a hook chapter. I think it's important that you wrote this to give yourself an idea of what you want to have happen in the first chapter, but, like i said, it feels more like an outline than the actual story.

I like the idea of this occurring over a decade so that there will be children from adolescent to toddler.
I like the detail in the video of the cats peeking in and meowing.
I don't think it was necessary to use the past perfect for such a large portion of the story. You might want to consider changing the tense to make it read smoother.