1) Yes it will. Anything that changes the fuel consumption, or power, will effect the carbon emissions. However, if its your vehicle tax you are worried about, it wont affect that, as that is based on manufacturers figures, not what your particular vehicle produces.
2) You will likely gain both, but not at the same time. You will have greater torque, so will use less energy to drive at the same speeds as you normally do. So if you drive exactly the same, you will save on fuel. However, you WILL boot it about a bit, as it does totally transform the car. So when you do this, you will use the same, or more, fuel than normal. Any potential savings in fuel are solely down to your driving style.
3) There are many places that will offer this service. You will likely be getting a generic remap, which means its basically the same file applied to many cars. This means that the map will likely be thoroughly tested on many different vehicles, and will be within safe tolerances. However, it wont maximise the performance of your particular vehicle. There are many different options for generic remaps, REVO being one, then Superchips, GIAC, Custom Code etc... will all likely provide you with generic maps. And these are most likely to suit your requirements. If you have other modifications, a custom map may be better suited, but it doesn't sound like that is required for you.
As NAMARK stated, even generic maps can come in a variety of flavours. My last car (A3 2.0 TDI) was mapped using a ST3 unit, which allowed me to switch maps at will. I had the choice of 3, Economy, Fast Road and another I cannot remember the name of just now as I only really had them 2 along with my original file.
There is, in general, a small difference in power (+/- 20 BHP) and torque (+/- 30 lbft) between the different maps, which may or may not be noticeable to you. I found the economy map to be faster than standard, and did me well on the economy front too, so stuck with that for a year. Then I stuck the fast road on it, and was more than impressed with that also, so left that on for a year or so. This option may not be available to you though due to the ECU in the newer TDI's. I know it wasn't possible for me anyway.
There is a lot of info out there, and as with everything, you will find good and bad reviews for any mapping companies. Your best bet is to look for someone local who do a generic map from one of the larger companies, REVO or Superchips for example. Then see if you can get a 14 day money back guarantee. If the map is good, they wont see you again until you start looking for even more power.