For the ones that do this (it's annoying) I go onto Google translate and listen to the sound (or look at the spelled sound) and then use my knowledge to write that in the Japanese characters we have been taught. It seems like this is tye nest way so you can get it right and not completely cheat. I did read that Duolingo should be used with other resources. It is annoyi g that they drop these questions ojt of the blue, though. Good luck!
That い is for indicating that あまい is an adjective, specifically an い-adjective. It's the part that gets conjugated when you need to change the adjective into other tenses.
甘い = "sweet", root/present tense
甘くない = "not sweet", negative tense
甘く = "sweetly", adverbial tense
甘かった = "was sweet", past tense
甘くなかった = "was not sweet", negative past tense