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I don't think its possible directly since Tata Sky uses Insat 4A and DishTV uses NSS6 satellite. It may be possible if you connect another dish with LNB through some diseqc switch to the receiver. Still, you would need their subscription card. Even then, I don't know if DishTV receiver will allow adding other frequencies even though TRAI has said that inter-operability must be maintained . But, why do you want to watch Tata Sky? Dish TV has more channels. Or do you want to be like pappe?
Regarding Dish DVR, they have already launched it a few months ago. But I heard that it's software was buggy. Don't know the present status. Check their
site.
Internet won't be viable economically and in terms of speed. Also, I doubt the reliability as well.