I'm pretty sure that the problem is a conflict with the spry tabs you are using. As a test of this you can try applying a single tab to the working page with the form but do not include the form in the tab. If you can get it working with the form and the tab on the same page the next test would be to include this form in the one tab and test it there.
If the form will just not work when inside of the tab then some extra tweaking may be in order if it is possible to get them to work together. Let us know about the results of your testing.
For the behavior you are experiencing this is as designed, you will need to place the cursor in the text area for the datepicker to show. As for a time picker this is not a control that I have ever made use of myself so I do not have any good suggestions for one. I'm sure that something can be found out there with a little searching though.