Learn Hardcore Korean: eoje nalssiga chuwosseoyo. - Yesterday’s weather was cold.

The particle 가 is the subject marker. It tells you that 날씨 (“weather”) is the subject of the sentence—i.e. the thing that is doing or experiencing the action/state (being cold).

Why is the verb ending -었어요 used in 추웠어요 instead of -아요 or -아요?

추웠어요 comes from the adjective (descriptive verb) 춥다 (“to be cold”). To make it past tense polite, you attach -었어요. Because 춥- ends in the vowel 우, you use -었- rather than -았- (vowel harmony rule).

Can I say 날씨가 어제 추웠어요 instead of 어제 날씨가 추웠어요?

Yes. Korean has a relatively free word order as long as you keep the verb at the end. Placing 어제 later emphasizes 어제, but both orders are grammatically correct:
• 어제 날씨가 추웠어요
• 날씨가 어제 추웠어요

What part of speech is 추웠어요? Is it a verb?