Not sure if you are interested, but here's what I found through youtube!
I found the pet websites you were looking for I think, archived in some fashion even if small.
Not a whole lot, but bits and pieces. While you can't directly pull jpgs from the videos, you
can relive the experience of the website from watching someone else play it or use it.
Granted these won't help with finding out why they died, but my surmise is being too similar to neopets and dying to the competition against neopets. I notice a lot of these youtube videos depict websites that completely mimic early neopets to a T. Marapets founded around the same time went a completely different route than neopets and thus was capable of surviving on it's own. Webtraffic would play a big role I think in a websites survival. This doesn't negate the possibility of internal drama within development. You could interview the boopets artist though and ask about their time working on the website. I find the topic of archiving internet history fascinating! It's fun to search for things in unconventional places!
To sympathize with your concerns, I grew up adoring flipnote and a lot of flipnote art/animation as become lost to the internet! Say, for the grace of youtube and its ease of access to kids uploading the things they loved back in the day. The quick rise and fall of a website too could be due to lack of financial gain (in flipnotes case) and obsoletion with modern webbrowsing.
Teripets: Teripets appears to be using a reworked version of neopets original code. (Neopets Classic Archive)
Misticpets: Wow! This is early 2000s design! I am astonished bits and pieces of it are still available to be seen on youtube! Appears to also be using something similar to neopets classic. I notice this with a lot of old websites. :0
Boopets - This is all I could find of it! But the artists behind the website appear to be currently active on youtube as of 2 weeks ago.