<p>Does excess use of AJAX affects performance? In context of big size web-applications, how do you handle AJAX requests to control asynchronous requests?</p>

AJAX itself (being asynchronous requests).. No not generally.
However if you have an abundance of javascript and markup and have large amounts of data transferred via your xmlhttprequests then yes you can see a performance hit.  It really depends on how you want your website to function any degredation is generally avoidable if sculpted correctly.