Learn Arabic (Modern Standard): المطر ينزل الآن.

Usually, Arabic sentences start with a verb (Verb-Subject). But here, the noun المطر (the rain) comes first. This is called topicalization. You use this Subject-Verb order when you want to draw special attention to the subject itself. Starting with the noun highlights that we are talking about the rain specifically.

Why is there no Arabic word for 'is' to make 'is falling'?

In Arabic, the present tense verb ينزل covers both 'falls' and 'is falling' all by itself. You don't need a separate helper verb for 'is'. The time word الآن (now) at the end makes it perfectly clear that the action is happening right at this moment.

If I wanted to just say 'Rain is falling' in general, could I remove the ال (the) from المطر?

Actually, no. In Arabic, when you put the subject at the very beginning of the sentence to make it the topic (Subject-Verb order), it almost always has to be definite. That means it requires the prefix ال (the) to be grammatically correct at the start of the sentence.

Does the word الآن (now) always have to go at the end of the sentence?

Time words like الآن are quite flexible in Arabic. Putting it at the end of the sentence is very natural and common, but you could also put it at the very beginning if you wanted to heavily emphasize that it is happening right now.