Congrats on your purchase and welcome to the club!
Regarding the tax, i suspect the dealer you got it from gave you the green slip intended for him - you may find you have to apply to the Divvy-La-La for a new V5 log book in your name.
However, before doing any of that, try on monday afternoon again using what you have - i've known the website often has problems over weekends, especially those at the end/start of a month.
As for your AC, silly question but you are aware there are two separate temperature controls, one for each side? If you set the passengers to minimum and then the drivers, if you don't touch the passengers you can control both from the drivers side.
Next option is that the DCCV is sticking so try running the temperature for the drivers side up to max and down to minimum while driving or sat with the idle speed raised to ~2000rpm to help circulate the coolant. If the temperature hasn't been changed in a while the DCCV can stick and you will only get hot air out, no matter what. (It's a useful preventive maintenance at least once a month to cycle the temperature like this - you don't want to have to change the DCCV!)
Alternatively, phone the DVLA tomorrow and ask them what's going on - they seem to be making it harder and harder to tax any vehicle these days, high time VED was scrapped and a small amoutn added to fuel duty to replace it - fairer and much easier to administer!