You are right about the 21 days - no visa required for British citizens.
Also your budget should be more than enough. To give you an idea - i stayed in the best hotel in Surigao (there are only 2) - it was comparable to £200.00 a night hotels here. I went daft in the bar, bought drinks, food and mango splits for everyone and their dog and at the end of the week my bill came to £88.00 yes its true!
Never been to Doha but since you are not leaving the airport, you never go through immigration so stay on international `soil` and wont need a visa.
You have to be careful anywhere in Filli, the people are wonderful and very friendly but there are a lot of robberies and everyone will know you are there and exactly where you will be going and when you will arrive - the Fillies are txt mad.