ame ga hutte mo gakkou ni ikimasu.

This is the て-form of a verb plus the particle も, which creates the “even if…” meaning.
• 降る (ふる) is an う-verb.
• To get the て-form, you replace the final う with って → 降って.
• Adding も gives 降っても, literally “even if it rains.”

The particle が here marks 雨 as the subject performing the action 降る (“to fall/rain”).
• 雨が降る is the standard way to say “it rains.”
• You wouldn’t use を because 雨 isn’t an object.
• You could use は (雨は降っても…) to contrast or emphasize “as for rain, even if it rains…,” but が is neutral and most common.

• に marks the destination of movement (“to school”).
• へ can also mark direction (“toward school”), but に is more specific about arriving there.
• を would mark an object being acted on, which doesn’t fit here.

• 行きます is the polite (ます-form).
• In casual speech, you could say 行く:
雨が降っても学校に行く。
• In very formal writing or announcements, you might choose 参ります instead, but that’s rare in everyday conversation.

Yes. Common casual versions include:
• Drop particles when context is clear:
雨降っても学校行く。
• Use contractions:
雨が降っても、学祭(がっさい) … (if you have a reason to shorten).
But the original polite form is good for general use.

• ～ても = “even if…” focuses on a concession (regardless of what happens).
• ～たら = “if/when…” describes a condition or time.
Example contrasts:
• 雨が降っても行きます。→ “Even if it rains, I’ll go.”
• 雨が降ったら行きます。→ “If/when it rains, I’ll go.”
The first stresses no matter what, the second just sets a condition or timing.

You can, but the nuance shifts:
• のに (even though): implies mild surprise or adversity.
雨が降っているのに学校に行きます。
(“Although it’s raining, I’m going to school.”)
• けれど (but): links two statements, softer contrast.
雨が降っていますけれど、学校に行きます。
(“It’s raining, but I’ll go to school.”)
Both are possible, but ～ても directly conveys the “even if” concession without implying surprise.