State (province) Not Showing!

It doesn't work because you have a javascript error on the page:
Uncaught RangeError: Maximum call stack size exceeded

Thats likely caused by the jquery javascript library used by your template which enters in conflict with mootools, the standard joomla library for javascript.

That is no simple to solve.
A work around you could do as you are in a hurry is to edit the state custom field via the menu Display->Custom field and display it as a text box so that users can just write their state. Or chang its type to dropdown and enter all the states from both countries manually.

It could be yes. Did you set the option "when a product is added to the cart" to "stay and display popup notice" ?
If so, then that's indeed the same cause.

Well, the problem is that mootools and jquery are not good friends. I would say that if you deactivate the dynamic menu at the top it might remove the errors. Or maybe configure your template to use a different display for the top menu...

If you changed the state dropdown that would explain why paypal is not happy with the address
The default zone type has a mechanism to use give to paypal the corresponding zone code (a 2 letters code). If you changed the type of the custom field, it means that the zones are not used anymore so the systems sends directly to paypal the value selected in the dropdown, which is then not recognized by paypal, thus the error.

So I would advise that you unpublish the address_state field and create your own custom field (which could have any values your want,s ot he same as you have now). By default you won't see your field in the address displayed on the invoices/emails, but you can change that by editing the files "address_template" via the menu Display->Views.

I don't see any paypal payment method on your checkout and I can see your state dropdown. So none of the problems described here apply to you. Please explain your problem, because I didn't see any on your checkout.

Yes, the country field should be published. But you can easily hide it which just one CSS line that you can add in the front end CSS file of HikaShop via the Display tab of the configuration:

.hikashop_registration_address_country_line{ display: none;}