The series with the doll burning in the doll's house was 'Tottie: the story of a doll's house'. It was an Oliver Postgate (of Ivor the Engine, Bagpuss etc. fame) production,based on a 1940s childrens' book called 'the dolls' house' by Rumer Godden. The book was re-published at the time of the tv series under the same title as the show.
There wasn't anything about a supermarket freezer in that one.
The doll that burned was called Birdie (made of celluloid, not wax, though a lonley wax doll at a doll exhibition did feature briefly), and got burned up when she jumped in between a little plush doll (Apple, who was her 'son' in the family of dolls) and a candle. It was all the fault of an evil porcelaine doll called Marchpane. Porcelain dolls still make me nervous.
The title character was a little wooden victorian doll who was the 'daughter' of the family.
It was a great show and I read the book, which I still have, afterwards.