I think what you are experiencing is a conflict of the js code because you have this content inside of a spry tab or some other js enabled area. If it working on the other page this indicates that it is the context of where you have it that is causing the problem. You may want to try testing it with another spry control to see if you get a better result.
What you are describing about the field not showing when you hover over it is as designed, you will only get the datepicker to show when you put your cursor inside of the date field. I don't really have any good suggestions for a time picker, it is not something that I have used myself. I'm sure that there are some controls out there though that you could manually incorporate into your form.
