On the Configure Site > Basic Setup > Contact Form section, you can customize 10 fields, and a select menu with 5 option.
If you need something more complex, you can use the UltimateWB Full Forms Builder app. Here is the article on how to use it:
How To Create And Integrate Forms & Surveys In Your Website
This will allow you to customize unlimited fields, set the field type, and assign drop downs. Including the "date" field, etc. It looks like this is what you would want to use.
David