No Address Identity in My Account

What do you mean by "idendity title" ? I can see the titles (Mr) on your screenshot.

Addresses can be used for billing or shipping. There is no type attached to them.
The billing address of my first order might be the shipping address of my second order...

It's either because:
You have a javascript error on your page which prevent the system to load the states of your country
You unpbulished/deleted the states of the UK
You removed the states from the subzones listing of the UK country zone.

That's indeed the problem. It means that you deleted the UK states from the database and the system cannot find them anymore.

To solve the problem, the easiest is to do like that:

Go in your PHPMyAdmin and empty the tables jos_hikashop_zone and jos_hikashop_zone_link.
Then, install HikaShop on top of your currently version.
That will put back all the zones with their default values and it should work again.

The UK states have been fixed in a previous release so if you install a recent install package you won't have to delete any UK state.

Did you empty BOTH jos_hikashop_zone and jos_hikashop_zone_link tables ?
It's just not possible that the blanks are still there if you emptied both and then reinstalled.

Could you give a link to your shop ?

So the dropdown of states works when you change the country. But the default dropdown does not display because you have a javascript error on your page:
Uncaught TypeError: Object #<Object> has no method 'getProperty'

That error is caused by the mod_superfish module you're using. If you remove that module it should work properly.
It's the problem described there: www.hikashop.com/en/support/documentatio...tibility-issues.html in the "problems with modal popups".

There is a documentation on how to configure the PayPal Pro plugin:
www.hikashop.com/en/support/documentatio...-paypalpro-form.html

The problem you're talking about is that the previous states you created yourself and the states that we have now in hikashop by default don't have the same namekey. Thus, the system cannot find them.
You should look at the namekey of the states in the jos_hikashop_zone table and then run the query below to update the state namekey in the existing addresses that your users already had with such query (to run in phpmyadmin):
UPDATE jos_hikashop_address SET address_state = 'NEW_NAMEKEY' WHERE state = 'OLD_NAMEKEY';

If you don't find an equivalent to a state you previously had you should add it in the zone management screen and then do that procedure so that the old namekey in addresses is replaced with the new one.