If the government wanted to do crypto-currency mining, I think they would set up a mining site by themselves, why would they want to subsidy you on electricity fees for you to do mining by yourself. Bitcoin price might goes up a lot if governments of a few countries are very active in mining bitcoins.
Not necessarily. Challenge being to run such an operation yourself, you'd have to find and hire the right people to maintain said operation. Governments being governments, this would come with an unnecessarily large administrative overhead, so in many cases it's more viable to simply provide the right incentives for private entrepreneurs. It's similar to how privately owned companies tend to run more efficiently and effectively than state run companies.