Translation:I climbed a mountain, and I am tired.
You are right, the usage is as you described, but the て-form for い-adjectives like あたたかい is not formed like the past form but with くて.
You drop the い and attach くて for the て-form.
warm: あたたかい -> あたたかくて
blue: あおい -> あおくて
Nouns and な-adjectives have olso a て-form. You can see a full explanation here: https://wp.stolaf.edu/japanese/grammar-index/genki-i-ii-grammar-index/te-form-joining-sentences-genki-i-chapter-7/
(Also it is something をかってください. The を is between the noun and the verb and not in front of the ください.)
-te form can be used to explain the chronological relationship between two clauses.
It's funny, everyone's comments on here are valid, they're just just not focusing on the flow Duolingo has. 'My dog sells hats' is also a "crazy" sentence, but i believe the point is how to regard these sentences to how it is being said than how it should be said, you knowよ。Besides, my real concern was 'I climbed 「the」 Mountain, and I'm tired'. Why can't i say "the" in this sentence?
'-mashita' or ta-form is generally used when you want to describe a physical / mental condition.
・ I feel __ (at that moment)
・ ____ appears to be ... (at that moment)
e.g., I'm so tired. (to be tired ... tsukare-masu)
(polite form )
(past tense of 'masu-form ')
Past comments on status verbs like 疲れる suggest 疲れました means became/ was tired in the past 疲れましています means became and still am tired continuing So the translation "am tired" seems incorrect & should be "I climbed the mountain and I was tired" ? O/w the 日本語 should be 山にのぼって疲れています ?
Those comments were not correct. I highly recommend reading Hitomi Hirayama's article "-mashita" is not a past tense! from her Pera Pera Penguin series in the Daily Yomiuri, which I have already quoted in a comment above.
My understanding of the difference between 疲れました (tsukaremashita) and 疲れています (tsukarete imasu) is not perfect, but to me 疲れました means I did something tiring, and now I'm tired. For 疲れています (tsukarete imasu), it feels to me like something has been going on for a long time that's making me tired. It's an ongoing state. Maybe if I climbed a mountain every day for a week, at the end of the week I would be 疲れています。