На березі річки вода шумить тихіше, ніж машини на дорозі.

Because на here means on / at in the sense of location, so Ukrainian uses the locative case.

• берег = bank, shore
• на березі = on the bank / at the shore

If there were movement onto the bank, you would more likely see the accusative:

• йти на берег = to go to the shore

But in your sentence, the water is located there, so на березі is locative.

Because this is a possessive relationship: the bank of the river.

In Ukrainian, of the river is usually shown with the genitive case:

• річка = river
• річки = of the river

So:

• на березі річки = on the bank of the river

This is a very common pattern:

• дах будинку = the roof of the house
• сторінка книги = the page of the book
• берег річки = the bank of the river

Because вода is the subject of the verb шумить.

• вода = water
• вода шумить = the water makes noise / the water sounds

Subjects are normally in the nominative case in Ukrainian.

Because the subject вода is singular.

• вода = singular
• so the verb is шумить = it шумить

If the subject were plural, the verb would also be plural:

• води шумлять = the waters are шумлять
• машини шумлять = the cars are noisy / make noise

Шуміти means to make noise, to rustle, to roar, to sound noisily, depending on context.

With water, it often suggests natural sound:

• flowing
• rushing
• splashing
• murmuring

So вода шумить is not just mechanical noise. It often has a natural-sound feel, like the sound of water being heard.

Because the sentence is making a comparison:

• тихо = quietly
• тихіше = more quietly / less loudly

The pattern is:

• X ... тихіше, ніж Y = X is quieter than Y / X sounds quieter than Y

So:

• вода шумить тихіше = the water sounds more quietly / more softly
• ніж машини = than the cars

In natural English, you would usually translate this more smoothly as the water is quieter than the cars.

Here it is functioning as an adverb, because it describes how the water is making noise:

• шумить як? = makes noise how?
• тихіше = more quietly

Compare:

• тиха вода = quiet water → adjective
• вода шумить тихо = the water sounds quietly → adverb
• вода шумить тихіше = the water sounds more quietly → comparative adverb

In practice, the form тихіше can look the same as the neuter comparative form, but here its role is adverbial.

Ніж means than in comparisons.

So:

• тихіше, ніж машини = quieter than the cars

This is one of the standard Ukrainian comparison patterns:

• більший, ніж... = bigger than...
• швидше, ніж... = faster than...
• тихіше, ніж... = quieter than...

In Ukrainian, a comparison introduced by ніж is normally separated by a comma.

So:

• вода шумить тихіше, ніж машини на дорозі

This is standard punctuation.

Because Ukrainian often omits a repeated verb when it is easy to understand.

The full idea is:

• вода шумить тихіше, ніж машини шумлять на дорозі

But repeating шумлять is unnecessary, so it is usually left out:

• ...ніж машини на дорозі

English does this too:

• The water is quieter than the cars (are).

Because машини is understood as the subject of the omitted verb:

• машини шумлять = the cars make noise

So even though the verb is left out, the grammar is still based on a full clause:

• ніж машини на дорозі (шумлять)

That is why машини stays in the nominative plural.

For the same reason as на березі: this is location, not motion.

• дорога = road
• на дорозі = on the road

Locative is used after на when something is located somewhere.

If there were motion toward the road, you might get accusative:

• вийти на дорогу = to come out onto the road

But here the cars are simply on the road, so на дорозі is locative.

Yes, Ukrainian word order is fairly flexible. The basic meaning would stay the same, but the emphasis can change.

Your sentence:

• На березі річки вода шумить тихіше, ніж машини на дорозі.

Other possible orders:

• Вода на березі річки шумить тихіше, ніж машини на дорозі.
• Вода шумить тихіше на березі річки, ніж машини на дорозі.
  (possible, but less natural depending on emphasis)

The original version sounds natural because it starts with the setting:

• На березі річки = on the riverbank

That gives the sentence a nice scene-setting structure.

Yes, very often.

Literally:

• на березі річки = on the bank of the river

But in natural English, this is often simply:

• by the river
• at the riverbank
• on the riverbank

So the literal structure is useful for grammar, but the most natural English translation may be less literal.

Yes, but the meaning changes slightly.

• на березі річки = on the riverbank, specifically at the bank
• біля річки = near the river, by the river

So на березі річки is a bit more specific. It places the water right at the bank/shore area, while біля річки is a broader near.

It can be understood either way, depending on context.

Ukrainian present tense often works for:

• a general observation
• something happening right now

So вода шумить can mean:

• the water makes that kind of sound there in general
• or the water is making that sound now

Without extra context, both are possible.