How to format a teaching statement?
In writing research and teaching statements, I generally use past tense for activities that have been completed, and present tense for ongoing activities—the same way you would in standard "ordinary writing."

Statements such as these do not need to be signed, as they are not letters. (Your cover letter for the whole application is a reinforcement of this.)

As for talking about the position, it certainly seems like a good idea to me to talk about how your experience and teaching philosophy fits the specific position in the application, doesn't it?