Regarding the prayers of a traveller:

1. If someone was born in a city, say Najaf, and they migrate to the UK permanently after marriage. However their parents still live in Najaf and they go back to visit occasionally around once a year or once in two years. Do they have to pray full when they go back to visit even if they visit for less than 10 days?

2. In this same situation, if they visit for less than 10 days, would they need to fast if it is Ramadhan?

3. If someone goes on holiday to a big city for more than 10 days, and say you intend to go out for sight seeing, if you intend travel more than 44km from your place of residence (hotel) within the city but do not leave the boundary of the city, does this count as you breaking the 10 day rule?

1- If he/ she still keep the intention that Najaf is his/ her home, then he/ she have to pray full, while, if he/ she -after migration- ignored Najaf as home city, and made the intention that he/ she will never go back to live in Najaf , then the prayer is qasr, if he/ she will not stay 10 days.
2- It depends on the intention as in number 1.
3- If you're referring to the big city, the sprawling supercity, like London, New York and Tehran, then yes.