The answer to the first problem is 66.
This is because this is the answer you get when you add 1+2+3 and on and on until you get to 11 is 66. (The first year he has one candle, the second he has two, the third he has three and so on). The quick way to do this is to add one to 11, multiply the two numbers together, and divide it by 12. So 11 x 12 divided by 2 is 66.
On his seventh birthday he would have used 28 candles. On his twelfth birthday 78 candles.
I estimated how many candles might be needed if each candle lasted two years instead and thought it might be 33, because that is half of 66. But actually it is 36 candles. I don't understand why yet but it does make an interesting pattern. Because each year we can use the candles again from the year before, the pattern goes like this:
1+1+2+2+3+3+4+4+5+5+6
So on his 8th birthday, they use the four new candles from the year before, and need four new ones. Then on his ninth birthday they can re-use four again, but this time they will need another five. I think it has something to do with odd and even numbers but I'm not sure.