Learn Arabic (Modern Standard): المطر ينزل في الخريف أيضا.

You are right, Arabic normally uses a Verb-Subject order (like ينزل المطر). When a sentence starts with the subject instead, like المطر ينزل, it is to emphasize or draw attention to that noun. It makes the sentence specifically about the rain. This is called topicalization.

Because the sentence starts with المطر, does it change how the verb ينزل works?

The verb still has to match the subject in gender. Since المطر is a masculine noun, the present tense verb ينزل (falls) starts with the masculine prefix يـ (ya-). If the subject were feminine, the verb would start with تـ (ta-).

Why does the word أيضا (also) end with an alif and two little diagonal lines?

Those two lines are called tanween fath, which makes an -an sound at the end of a word. This gives it the pronunciation aydan. In Arabic, many adverbs are formed by adding this -an ending, and words like أيضا typically sit at the very end of the sentence.

Does the preposition في (in) affect the noun الخريف (the autumn) after it?

Yes! As a quick refresher, prepositions like في cause the noun immediately following them to take a kasra (a short i sound) on its last letter. If you were reading this with full grammatical endings, it would be pronounced fi al-kharifi.