"저의 방이 좁습니다."
Translation:My room is narrow.
좁다 means the area or the width is small, not the overall size. If a house is very big, but 99% of this crazy house consists of walls and the inside is actually very small, the house is 크다 and 좁다. A road is usually 좁다, not 작다. The antonym of 좁다 is 넓다.
my guess is 좁은 means narrow, small in space matter ~ while 작은 means small . like small hat
제 is just a commonly used shorter version of 저의. Duolingo uses the "proper" way to write it, not slang.