I’m quite sure a lot of people out there is meeting problems with the latest edition (probably legacy too) of Contact Form 7 and the need to have date picking / calendar.

I recommend you to just get this one: http://wordpress.org/extend/plugins/contact-form-7-datepicker/

But warning first, because of the change in contact form 7, the settings for the datepicker plugin is now hidden/disappeared. So you cant customise it (unless you know how to hack it out).

So, after all the set up and stuff, I realised that the form does not work in Internet Explorer (damn them right!?), it works on all other browsers. Basically, at least the problem I met is that the submit button does not work.

So when you open the developer tool in IE10, you will see that the POST is done in text/html via XMLHttpRequest <--- which what you need is actually text/json via JS Library XMLHttpRequest. While this sounds complicated a thing to solve... however since I already solved it, I shall give you the easy answer: the problem lies in your doctype and an instruction for IE to use the correct version of document mode to open your website/webpage. Just change your top declarations to these:
To use the latest set of standards

and



To force the internet explorer to open the website with the stable and tested IE9 Document Mode.

There! And your contact form 7 should be working fine!