"I climbed a mountain, and I am tired."
"[...] and got tired" would be more accurate. 疲れる means "to get tired"/"to tire".
(As for myself) "Mountain climb [te-form]" "became tired"
Am I thinking about this structure correctly? The te-form is used because there's more to say in the sentence and it inherits the tense (in this case, past-tense) of the verb at the end?
Not sure if I understood your questions, but it is "I climbed mountain" then/as the result/because of that "I got tired." If you break down, it will be: 山にのぼった。そして(その結果・そのため）、つかれた。
It's used to connect verbs (and adjectives) in a similar way to 'and' in English.
I think it helps to consider how it's used in many connective contexts. Like "-te-kudasai" for "please ___".
Why "山に" instead of "山を"? Shouldn't it be a direct object for the verb "to climb" instead of indirect object?
I'm learning so I'll ask the more experienced to correct me where I'm wrong. As I understand it, "noboru" is an intransitive verb, which is to say it doesn't take a direct object. You aren't climbing something, you are just climbing. You are ascending, or going up. Where is that happening? The mountain, hence "ni".