I like number 4 best because it states the "why" right up front. I think there is too much elaboration, keep it short and sweet. Stress good credit score, long term rental & personal references and stable employment history. I would remove the word bulldog, that breed reference can turn some people off. It would be better to refer to the dogs as 2 professionally trained rescue dogs under _____ lbs.
Here is a possible draft that is more direct:
Hello
My name is Cordelia, I am a mature, young professional woman looking to relocate to a quiet and peaceful area. I'm originally from a farm in Scandinavia and want to get back to nature. I have been living in San Diego, California for the past four years and I have excellent long term rental and personal references (provided below). I am well employed, with great credit and ample cash reserves. I am looking for a long term home for me and my 2 professionally trained rescue dogs under _____ lbs. I do not smoke, drink or use drugs and am ready to put down a deposit today. I look forward to hearing back from you!
Thank you!
Regards
Cordelia