usa to uk travel help?

i am planning to meet my cousin as a surprise visit in may
he lives in hastings uk near train station.

1. how to travel from heathrow to hastings (i couldnt drive there)
2. how to travel from gatwick to hastings (is it easier and cheaper to travel by bus or train)
3.also in london nearest to tourist destination apart from hotels
do they have like a studio apartment for 14 days. i checked hotels cost 1000dollars for 14 days
3. can a us citizen travel from us to canada by land then take a flight from toronto to london. then return from london to canada by flight and canada to us by land.

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It is simpler to travel by train from Gatwick to Hastings. Southern trains operate an hourly direct service from Gatwick Airport to Hastings; the journey time is about 1 hour 20 minutes. From Heathrow you would need to catch the Underground to Victoria or Charing Cross Station in central London and board the Hastings train there.