Hi Hanuman,
1. He was a most unhappy man. Correct. One meaning of 'most' is 'very'; it is not a superlative here.
2. I am six years old next Sunday. OK. Present can be used for near future which is planned or assured.
3.Tomorrow will be Sunday. OK
4. We were having a car in those day. Nope. 'We had a car in those days.' 'Ha