How to address an Editor if I have new proof techniques of old results?
What makes this question challenging to answer is that it’s addressing the wrong level of generality: you’re asking about a very specific situation (publishing a simpler proof of a known result) but your questions suggest you’re missing basic facts about publishing of math research in general (how to “address an editor”, how to select a journal to send your paper to, and how to know if your research is even publishable) that have little to do with the specific situation you’re describing.

In terms of answering these questions, only one strikes me as answerable with the information you’ve provided: in math you don’t “address an editor” in any special way, you simply send in your paper using the process described on the journal’s website. If this involves an email to the editor, this need not contain anything more than a factual statement that you are submitting your paper to the journal. There is no need to “address” the contents of your paper or attempt to support the paper’s case for publication in the email - that’s the job of the paper itself.

For the other questions, I think you’ll have to find a knowledgeable person to actually look at your paper and give you advice about whether the results are publishable and what journal they will be a good fit for, how to choose a good title, etc. Those are all important questions but they would depend on much more detailed information than the mere fact that you’ve found new proofs of known results - papers meeting such a description can range anywhere from near-worthless to something that belongs in a top journal. Anyway, good luck!