1. Not sure. What's the power consumption without the battery charging?
2. Not a transformer necessarily, but something like this "
converter". A transformer needs a certain minimum load to operate effectively and can operate poorly with less than this load. A converter is usually rated for a smaller load (50 watts or less). There are switchable converters that combine a converter and a transformer.
Or you could use the 120v shaver outlet in the bathroom. I've used the shaver outlet to charge my camera batteries when all I had was a 120v charger.
3. For the UK all you really need is the big adapter in the back of the picture. You can get just that (not at RS) for a few bucks; the RS price is OK for the whole set.