I am sure enjoying the theatre
of this thread. It certainly is very colourful
. I'm hoping the responses stay civilised
and people don't continue to criticise
spelling. I also hope no one takes offence
at my attempt at humour
To answer the OP's questions...
As stated the TC4 uses 12mm wheel hexes, which is considered standard for 1/10 scale 4wd touring cars.
There are many different companies that make 1/10 wheels & tires. You can find a wide variety of both from companies like HPI Racing, Tamiya or even Team Associated. However there are also many different companies that sell wheels/tires. Sometimes availability can be limited based on location around the world.
As a general rule, just about any foam or rubber tire will work well enough on rough concrete regardless of temperature. Concrete that has a broom finish such as the picture below will have plenty of grip with very little prep (other than blowing off dust/debris)
It can be more challenging to choose the right tire for tarmac/asphalt. Depending on the age of the surface, temperature, or if a sealant has been used you may a different tire to achieve desired grip levels. To improve traction many clubs will use a sugar water solution sprayed onto the surface. A quick search on RCTECH will show various way to make/apply sugar water including hints on keeping bugs away.
Heavily varnished (wood or concrete) floors can be the hardest to achieve the desired level of grip. Since the surface tends to be very smooth, any amount of dust will be like driving on an ice rink (but it is great for drifting). Since these type of surfaces are typically found indoors, the owners typically frown on applying additives to the floor to improve grip. Here is an old thread on RCTECH that might be useful (What tires to use on Polished Wood Floors
One final thought. Since you are located in the UK, I would recommend seeking out a local club to ask for advice. You might start in the UK RACING forums here on RCTECH (http://www.rctech.net/forum/uk-racing-136/
). Luckily people in the UK race 1/10 touring cars on just about any surface and weather (including rain) so someone near you might have the answers you are looking for. They also might have some used/spare tires sitting around for you to try (hopefully for cheap/free).
Hope this info helps.