1. After the passing of George Rose's 1812 Act, parishes were supposed to use printed registers to record baptisms, marriages and burials, and in the case of a marriage the occupation of the groom was supposed to be included. However as far as the exact wording is concerned, it would probably depend on what he told the priest and what the priest decided to record. This might be as short as 'soldier' or 'trooper' or might include his regiment.
2. As far as I can see his service record does not appear to have survived. If he had died while still serving, it was normal to destroy a soldier's record as it was no longer required for pension purposes. If it had survived, his wife might be recorded if she was formally recognised by the Regiment. This meant that she was allowed to live in the barracks with her husband (and children) and to be moved as part of the regiment when it deployed to other stations overseas. The children are unlikely to have been recorded.
3. The 3rd Dragoon Guards were deployed as follows over the period:
June 1816 Ireland
1829 England
1834 Ireland
1837 England
1843 Ireland
They don't appear to have gone to India until August 1857 following the Indian Mutiny. However as Shaun has said he is recorded as being with the 3rd Light Dragoons in Karaul, Bengal in April 1841.
Also as Shaun says, the best source for exactly where the regiment and its detachments were positioned would be the the pay and muster rolls held at the National Archives. These were compiled monthly and usually show exactly where each individual soldier was. They would also provide clues as to whether his wife was 'on the Regiment' (ie officially recognised) because of the pay and allowances William would have been paid. The rolls have not been digitised and so are not available online. You would need to visit TNA or arrange for someone to examine the relevant entries for you. The muster rolls are in the following series: WO12 / 207 - 217, covering the period 1823 -1847.
4. He was buried on 29 April 1844 in Kussowlie, Bengal.
