"A dog is climbing a tree."
It's just one of those things you've got to remember. にtypically goes with のぼる. I haven't seen を with it.
Also just a little fun fact, 登る and 上る are both read as のぼる but 上る is more like a gradual ascent like a small hill and 登る is for mountains and trees. For stairs you can also use 上がる(あがる).
Yes and no. I completely agree that in this exercise, it IS pronounced "KI" and not "MOKU" (Sorry, I don't have the Japanese keyboard activated on this laptop so it's Romaji)... MOKU does mean tree, but it is the on-yomi reading and is only used when combined with other Kanji, such as "MOKUYOUBI" (Thursday). But when it is by itself, it is "KI" as it is in this sentence. I'm glad someone else noticed that.
I've also seen "CHUU" and "NAKA" used in the matching exercise earlier in the course, but the problem there wasn't the audio - but the match was CHUU instead of NAKA (i.e., the voice said "NAKA" but the Hiragana it was matched to was "CHUU"). Again, confusion between On-yomi and Kun-yomi readings of the Kanji.
I think this is a really new class, but it doesn't look like they've beta tested it enough...
を is almost NEVER used with verbs that are considered 'directional movement' verbs. Go to, come to, go up, climb up etc. Are a few of these verbs. They almost always take に or へ. If を is used, for instance, as we've seen before in another lesson 'を行きます' it has a special meaning! It's not just there because 行くis a verb!