Learn Spanish (Latin America): ¿Cuánto tiempo tenemos hoy?

The word for "how much" or "how many" always has to match the gender and number of the noun it describes. Since tiempo (time) is a masculine, singular word, we use the masculine, singular form cuánto.

Doesn't the word tiempo usually mean "weather"?

It means both! Tiempo is the word for both "weather" and "time" in Spanish. You can easily tell which one it means from the context. Since we are asking how much of it we "have" (tenemos) today, it clearly means "time" here.

Where is the word for "do" in this question?

Spanish doesn't use helping verbs like "do" or "does" to ask questions. The verb tenemos simply means "we have", and just by changing your tone or adding question marks, it automatically translates to "do we have".

Would it be wrong to include nosotros (we) in this sentence?

Not at all! You could say ¿Cuánto tiempo tenemos nosotros hoy? However, because the ending of tenemos already makes it perfectly clear that "we" are the ones having the time, native speakers almost always leave the nosotros out.