What is the significance of this form?
It's not simply a round, chubby cat, it's a Maneki-neko (lit. beckoning cat). Maneki-neko is a common figurine in Japan. Shops have it displayed to beckons customers and wish bringing more fortune in.
Did he purposely choose this appearance?
Yes and No.
Nyankon-sensei was sealed in Maneki-neko for a long time before Takashi release it by accident, it ordinary takes on the shape and just be that way.
Ordinary people can see it when it is in the Maneki-neko form but not in the original Madara form. Still, we don't know if Nyankon-sensei like its form so much or on other purposes.
Why does Takashi call Madara by the name "Nyanko-sensei?"
Nyankon-sensei requests Takashi to call him sensei and as an exchange, it becomes his bodyguard.
Wikipedia actually has a well-written article about the anime & manga including characters background, in case you want more details.