"He is not late."
Yes, but from what I'm learning from my teachers, the "he" would normally get implied instead of stated, or a name would get used, instead of the phrase かれ In fact, all three of my teachers have explicitly forbidden the classes from using かれ or かのじょう so I'm curious as well as to why Duolingo insists on using them.
No, because osoi is an 'i' or true adjective. Ja nai is the negative present plain form of desu. The endings of adjectives like osoi inflect (or change) to show positive, negative, present, past - not unlike verbs. However the endings of na adjectives like kirei and shizuka don't change these adjectives stay the same and the plain form of desu shows positive, negative, etc for them like the endings of true adjectives do. Osoi is a true adjective so it doesn't need "help" from desu. Hope I've explained it well.
Since you are obviously an advanced student this might interest you. My friend, native Japanese, said that Duo's "correct" answer sounds very od!. Such a statement in English would not be used in the present tense. It seems to be a different way of thinking in.Japanese than in English. He said in Japanese it would have to be stated in the past or the future tense, never in the present.I am bringing this up for advanced students. If you are a beginner, please ignore this comment until later. I don't want to cause any more confusion.