Sne er lavet af små iskrystaller.

Sne means snow in a general sense, as a substance. Danish often uses a bare noun like this for general statements, especially with mass nouns.

• Sne er koldt. = Snow is cold.
• Sneen er kold. = The snow is cold.

So in your sentence, Sne means snow in general, not a specific patch of snow.

Er lavet af means is made of.

It breaks down like this:

• er = is/are
• lavet = made
• af = of/from

So the pattern X er lavet af Y means X is made of Y.

Example:

• Bordet er lavet af træ. = The table is made of wood.

Yes. Lavet is the past participle of lave, which usually means to make or to do.

So:

• at lave = to make / to do
• lavet = made

In er lavet af, it is part of a very common construction meaning is made of.

Because af is the normal preposition used for what something is made of.

• lavet af træ = made of wood
• lavet af metal = made of metal

Fra usually means from in the sense of origin, source, or place:

• Han kommer fra Danmark. = He comes from Denmark.

So af is about material or composition here, not geographical origin.

Because Danish indefinite plural nouns normally do not use an article.

So:

• en iskrystal = an ice crystal
• iskrystaller = ice crystals
• små iskrystaller = small ice crystals

If it were definite, you would use an article:

• de små iskrystaller = the small ice crystals

Små is the plural form of lille, meaning small/little.

Compare:

• en lille iskrystal = a small ice crystal
• små iskrystaller = small ice crystals

So the adjective changes because iskrystaller is plural.

Because Danish makes compound nouns as one word much more often than English does.

• is = ice
• krystaller = crystals
• iskrystaller = ice crystals

This is very normal in Danish. Other examples:

• kaffekop = coffee cup
• togstation = train station
• sommerferie = summer holiday

The last part is the main noun, and the first part describes it.

It is understandable and grammatical, but many Danes would probably say:

• Sne består af små iskrystaller.

That can sound a bit more natural or scientific when talking about what snow consists of.

Er lavet af is especially common with man-made things, such as:

• Stolen er lavet af træ. = The chair is made of wood.

So your sentence is fine for learning, but består af is often the more natural choice in this kind of factual statement.

Yes, but it changes the meaning slightly.

• Sne er lavet af små iskrystaller. = Snow is made of small ice crystals.
  (general statement)
• Sneen er lavet af små iskrystaller. = The snow is made of small ice crystals.
  (specific snow)

So sneen refers to particular snow, while sne refers to snow in general.