Date picker calendar icon missing after update 1.6.2

This is usually caused by a javascript problem. Hard to tell without being able to see the form first hand. If you can post a link to a page containing the form we can take a look and see what the issue is.

Taking a quick look at your form it seems there are a few issues and they are probably all being caused by a theme or plugin conflict which are preventing the gravity forms css and js files from being included on your form page. Your menu isn't working on the form page yet it is on other pages, the datepicker field isn't actually being activated so it is in effect remaining a standard input and the advanced ui isn't being applied to the program name drop down field.

You say that it was working a few weeks back, have you activated any plugins or a new theme since then, if so I would try temporarily de-activating them and then checking your form to see if the issues are still present. If you are still having problems you should work though the steps on the following page
http://www.gravityhelp.com/documentation/page/Testing_for_a_Theme/Plugin_Conflict

Thanks for the update