Alexa ranking don't mean squat when it comes to ranking web pages in the search engines, doesn't really matter what search engines your "target" ("target" quoted because targeting web pages for specific search engines is almost an obsolete SEO strategy).
There shouldn't be any significant considerations given to Alexa when deciding how to optimize webpages. Alexa can provide indicators as to what's happening on the web but they too are not very significant in the bigger scope of winning keyphrase competitions.
That is absolutely correct. The Alexa rankings are not to be trusted as sometimes the stats could be funny. For example, after experiencing some great Google rankings boosts, Alexa shows some traffic drops for your website but you do know that it is wrong.
When you do SEO, you're doing for all. It's just that different search engines have got specific algorithms. Yahoo, MSN tends to prefer sites with keywords inside their domain. Baidu and Yandex are pretty similar to Google's algorithm.
I have now seen an increase in traffic from Bing after optimizing it... Actually I have more traffic from Bing than Google right now and I like it alot! Make your site user friendly... that always works well!
Here are some tips for optimizing your site for Bing, Baidu and other search engines: