You can put fields in front or in back of other fields just fine in the form designer, just drag them on top of each other. However, keep in mind that only 1 field (that with the lowest Control ID of the two, ie the one in front) will be visible to the user. I would not recommend doing this as it can easily get confusing if you were to revisit your project later on to make changes and you don't remember exactly which fields are in front of each other.
Yes, the DateTime field needs to be visible to write back as I understand. The operator really doesn't need to see this and it takes up space. I was messing around with trying to get it behind the Done button and somehow may have gotten it behind the background or tab. It doesn't show up in the designer now, see desktop.jpg, but it does show up on the device in the front of Done button (see mobile.jpg).
I tried to play around with changing the control ids a bit, without luck.
It looks like you're making things unnecessarily complicated If you don't need a field to be visible to the user, you can just uncheck the 'visible' checkbox in the field settings. It will then not be created on the form when you set up a default form, and it will not be visible to the user, even if you do place its control on the form. In this case you can just go ahead and delete the field control from the form, this will not affect the field functionality, ie. the date will be still be recorded. Deleting a field from the form, does not delete the field itself.
If for some reason you don't need a field to be visible to the user, but you do need to have it on the form (for example if you use a Form Logic rule to refresh a specific control ID), then you can just drag it outside of the visible form (the area marked with the dotted line in the form designer).