The computer-generated model used for the Dimetrodon hatchlings is actually drastically smaller version of the fully grown Dimetrodon animation, with apparently the same body proportions and color-scheme. So when one of them (the newly hatched animals) is shown climbing up a tree, the viewer is treated to a rather awkward sight, since it looks just like an adult, to which of course it wouldn't be possible to climb up a tree.
The amphibian that threatens the Dimetrodon nest, and eventually gets eaten by one of those creatures, is never named in the show. It is a Seymouria.
The giant spider that appears in the first segment is called a "mesothelae", which is not the name of a genus or species, but the name of a broad group of spiders. Originally, this animal was meant to be a Megarachne, thought to be the largest spider ever discovered. But during the series' production, it turned out that "Megarachne" wasn't a spider at all, but a sea scorpion. Since production was already underway, the spider couldn't be removed from the episode, so the creators changed its name and pretended it's an as-of-yet undiscovered species. In reality, there is no evidence of any spider being as large as the one depicted in this program.