Mater vult tegulas novas supra fornicem poni ante hiemem.

The sentence breaks into two parts:

• Mater vult = Mother wants
• tegulas novas supra fornicem poni ante hiemem = an infinitive clause, the new tiles to be placed above/on the arch before winter

So the main verb is vult, and everything after it tells what the mother wants.

Poni is the present passive infinitive of pono.

• pono = I place / put
• ponere = to place / to put
• poni = to be placed

It is passive because the tiles are not doing the placing; they are the things being placed.

So:

• tegulas ... poni = the tiles to be placed

not

• tegulas ... ponere = to place the tiles

This is a very common Latin pattern. After a verb like vult, Latin can use an accusative + infinitive construction.

Here, tegulas is the subject of the infinitive poni, but in this construction that subject goes into the accusative.

So:

• Mater vult tegulas poni = Mother wants the tiles to be placed

Even though the tiles are the ones being placed, Latin still uses the accusative because they belong to the infinitive clause after vult.

Because they are not the subject of a finite verb. They are the subject of an infinitive.

Compare:

• Tegulae novae ponuntur. = The new tiles are being placed.
  Here tegulae novae is nominative because ponuntur is a finite verb.

But in your sentence:

• Mater vult tegulas novas poni. = Mother wants the new tiles to be placed.

Here tegulas novas is accusative because Latin uses the accusative for the subject of the infinitive after vult.

Latin adjective order is much freer than English word order.

• tegulas novas
• novas tegulas

Both can mean new tiles.

What matters is agreement, not position.
Novas agrees with tegulas in:

• gender: feminine
• number: plural
• case: accusative

So tegulas novas simply means new tiles.

Because it is the object of the preposition supra.

• supra = above, over, on top of
• supra normally takes the accusative

So:

• supra fornicem = above the arch / on top of the arch

Also, fornicem is the accusative singular of fornix.

It tells the place where the action of poni happens.

So the phrase answers the question:

• Where are the new tiles to be placed?
• Supra fornicem = above/on the arch

It modifies poni, not vult.

In other words, the mother does not want above the arch; she wants the tiles to be placed above the arch.

Because ante is a preposition that takes the accusative.

• ante hiemem = before winter

This is just like:

• ante bellum = before the war
• ante lucem = before dawn

So hiemem is accusative singular after ante.

Not in the same way English does.

English says:

• Mother wants the new tiles to be placed...
• or Mother wants that the new tiles be placed... (less natural in modern English)

Latin often uses an infinitive construction instead:

• Mater vult tegulas novas ... poni

So Latin does not need a separate word for English to here. The infinitive poni already carries that idea.

No. The form poni makes it clearly passive:

• poni = to be placed

If the meaning were Mother wants to place the new tiles, Latin would use the active infinitive:

• Mater vult tegulas novas supra fornicem ponere.

So your sentence means that the mother wants the tiles to be placed, not that she herself necessarily wants to place them.

Latin word order is much more flexible than English word order because the endings show the grammatical relationships.

English depends heavily on order:

• Mother wants the new tiles to be placed before winter.

Latin can move pieces around more freely:

• Mater vult tegulas novas supra fornicem poni ante hiemem.
• Mater ante hiemem vult tegulas novas supra fornicem poni.
• Tegulas novas mater vult supra fornicem poni ante hiemem.

These can all express basically the same idea, though the emphasis may shift.

So the order in your sentence is normal Latin, even if it feels unusual from an English perspective.

Yes:

• Mater — noun, nominative singular, mother; the subject of vult
• vult — verb, 3rd person singular present active indicative of volo; wants
• tegulas — noun, accusative plural feminine of tegula; the subject of the infinitive poni
• novas — adjective, accusative plural feminine of novus; agrees with tegulas
• supra — preposition, above / over / on top of, taking the accusative
• fornicem — noun, accusative singular of fornix; object of supra
• poni — present passive infinitive of pono; to be placed
• ante — preposition, before, taking the accusative
• hiemem — noun, accusative singular of hiems; object of ante

So the whole grammar is very compact, but each part has a clear role.