Is it possible to have 2 delivery periods in a day?

Yes, you can create two time slots in the day and set up a different price for them if you want to.

Can you show the delivery date as a field by default instead of the link?

Yes, it is possible by setting the related option in YITH Plugins->Delivery Date->Settings-> Show DatePicker. 

Is it possible to set the delivery date and time box as mandatory, so that customers cannot overlook it?

Yes, you can do that from WooCommerce Shipping settings. Select the order shipping method and then make sure you tick the mandatory checkbox to make the field required.

Mandatory delivery date