why dont we heat it and generate electricity in the house with the same source?
This is gradually being done more and more.
The problem is that electricity infrastructure is already there - connect a generator the grid and connect your customers to the grid and the electricity will flow (it's more complicated than that, obviously, but the delivery mechanism is already there)
If you want to transfer heat, you need to get big pipes which carry hot water from the power plant to the customers and cold water back again. Installing and maintaining buried insulated pipes is expensive and time consuming.
However, this concept of 'combined heat and power' is starting to take off, because of potentially large cost savings by using the waste heat.
On a small scale, you can natural gas furnaces, that are actually generators, and the waste heat from the generator is used for heating - these can be installed in place of a normal furnace. Essentially, you get 'free' heat when the generator is switched on. (These aren't partiuclarly popular yet, because the stirling generator mechanisms are extremely expensive, and not particularly reliable).
On a larger scale, places like hospitals have started installing systems like this - instead of using a gas boiler and a diesel backup generator - some have now installed gas-turbine generators which run 24/7 - cheap heat, cheap electricity and a backup power source, all in one.
On a larger scale, there are a number of municipal power plants, where the heat is piped to several thousand homes in the near vicinity. In some cases, these power plants are powered by refuse - in an attempt to be even more 'green'.