You say that you "would like to build."
Is that because you think you have a need for something other than what you have? Or that you're looking for a learning experience?
Please load photos directly to this site - click the paper clip button at the bottom left of the text box you type in for each post. I don't do DropBox.
The power supply in your first post is a flyback type, high frequency switch mode supply. It does use a transformer, although in a flyback converter it's more accurately called a coupled inductor.
But that transformer can only be used as part of a circuit like the one that it's in. You couldn't use it as a normal line frequency transformer.