Am I able to make the Date widget where it automatically goes to the next box instead of having to press tab?

When you insert the date of birth it isn't as seamless as I would like. You either have to physically type in the divider or press tab if its expanded. 

Similarly, am I able to put a default field like $ in a text box and it not be erased when the form participant is filling it out from one field to the next? Sorry I know this sounds vague but I am having difficulty explaining what I am referring to.

Ex: Question - What is your salary 

      Answer - $50,000 

I want the dollar sign to just be permanent and not able to be erased. Is this doable?

I'm not sure if I understood your requirement, the date picker automatically adds the separator when the date is selected, here is a sample form with some date pickers: https://form.jotform.com/73244774640964 

You will find more date picker widgets here: https://widgets.jotform.com/category/pickers 

Now, regarding the dollar sign, you may use the Masked Input widget: https://widgets.jotform.com/widget/masked_input 

If you're performing calculations then you may only add the dollar sign to the calculation. 

I hope this helps.