Weather is an uncountable noun, and the weather is a singular noun? Does weather plus the definite article the make it a singular noun? Please explain, thank you!

Weather, weather, is indeed an uncountable noun.

In English grammar, whether a noun is singular or plural has nothing to do with whether it is preceded by the or not.

The as a definite article, is generally used in specific situations. In addition, please remember that any noun in English (except for proper nouns such as names of people and places) cannot appear alone in the singular, that is, it must be preceded by the, a (only countable nouns can be added after a), that, this, or plural form appears.

In the example you gave, the "the" in "The weather" refers specifically to the weather in Taipei; and the "that" behind it is a pronoun, referring to "the weather".

You must learn grammar carefully. The exam is useful, although communicating with foreigners in the future will not be so rigid. . .