2nd show: Definitely mid to late 80s. A puppet show. No humans involved. Not Sesame Street. I only ever saw the one episode. The puppets were learning things about math. I remember a scene of a puppet singing home on the range or a parody of it.
I think you answered a different show I forgot about without knowing it! I googled Peppermint Park and found a video of dinosaurs. And those dinosaurs behind the news desk sparked a memory of a vhs tape my aunt had me see!
It also made me remember the show I saw had more cutesy/human looking puppets along the lines of the Muppets more human looking characters. Not flesh colored. Thanks for sparking a memory for me. bnr]