At last, some vocabulary I recognise from my school French lessons some 40 years ago. Weather seemed quite popular in those days. Funnily enough, not only did we not have much dying to worry about but I also don't remember many clothes getting mentioned. I remember hat (chapeau) but none of the others so far. I'm wondering how we filled five years of study without mentioning food, drink or clothing!
I did up to A-Level French and before that, our pre-GCSE and GCSE French skills were rather holey.
When we did A-Level French, we went to a school where their GCSE students and below were more involved and had a higher grasp than my school's equivalent. But throughout the whole seven years, not once do I recall hearing about a chapeau or manteau, clothes weren't touched on much.
We certainly entered A-Level looking rather uncultivated!
Your textbook is not wrong. When talking about weather conditions, you may say "il fait chaud", "il fait "froid". But when it is snowing, on dit "il neige" and when it rains, on dit "il pleut". https://www.thoughtco.com/french-weather-vocabulary-le-temps-1371465
French has no neutral "it". Everything is either "il" (he/it in English) or "elle" (she/it in English).
French is like English in that its grammar requires an overt subject, even if there really isn't anything making the verb happen.
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That would be more along the lines of "This is rain."
"Il pleut" is what's known as the impersonal construction. Also, don't let "il=he/elle=she" throw you off. French really does not have a neuter "it". Depending on grammatical gender, everything, even inanimate objects, are "il" or "elle" where in English we would say "it".
Because "pleut" is already a verb: pleuvoir to rain, pleut it rains/it is raining.
The noun "rain" is "pluie".
Also, rain and snow are weather events that have their own verbs. The "il fait ..." construction is for things like temperature or general quality where you fill in the blank with a noun or an adjective.