Сегодня курс в банке ниже, чем вчера.

Because курс is the subject of the sentence — the thing being described.

In Сегодня курс в банке ниже, чем вчера, the sentence is saying something about the rate. That makes курс nominative singular.

A very literal structure is:

Today the rate in the bank is lower than yesterday.

In Russian, the subject usually appears in the nominative unless something else in the sentence changes that.

Russian normally omits the present-tense form of to be.

So where English says:

The rate is lower

Russian simply says:

Курс ниже

This is completely normal. In the present tense, Russian usually does not use a separate word for is / am / are.

Compare:

• Он дома = He is at home
• Курс ниже = The rate is lower

В банке is in the prepositional case because it follows в in the meaning of in / at a place.

• dictionary form: банк
• prepositional singular: в банке

So:

• в банке = in the bank / at the bank

This is a standard pattern:

• в магазине = in the store
• в городе = in the city
• в банке = in the bank

Ниже is the comparative form of низкий and means lower.

Russian often uses a short comparative form like this after the subject:

• высокий → выше = higher
• низкий → ниже = lower
• дорогой → дороже = more expensive
• дешёвый → дешевле = cheaper

So:

• Курс ниже = The rate is lower

You can sometimes build comparisons with более plus an adjective, but with common comparatives like выше, ниже, лучше, хуже, the simple comparative is much more natural.

Чем is the normal word used for than in comparisons.

So:

• ниже, чем вчера = lower than yesterday
• лучше, чем раньше = better than before
• дороже, чем в прошлом году = more expensive than last year

It introduces the second part of the comparison.

Russian often leaves out repeated words when they are obvious from context.

So чем вчера really means something like:

• than it was yesterday
• than yesterday's rate
• than yesterday

Russian does not need to repeat курс or add a verb here.

English often does the same:

• Prices are higher than yesterday
• He feels better than before

So чем вчера is a very natural, compact comparison.

Yes. That is also natural.

Russian word order is flexible, and moving в банке changes the focus a little, not the core meaning.

Compare:

• Сегодня курс в банке ниже, чем вчера
  Focus starts with today, then introduces the rate in the bank

• Сегодня в банке курс ниже, чем вчера
  Slightly more emphasis on in the bank

Both are grammatical. Russian often uses word order to manage emphasis and flow rather than strict sentence structure.

Putting сегодня first highlights the time frame: today.

Russian often puts the most important or most topical information near the beginning of the sentence.

So this version sounds like:

As for today, the rate in the bank is lower than yesterday.

If you moved things around, the emphasis could shift slightly:

• Сегодня курс в банке ниже, чем вчера — emphasis on today
• Курс в банке сегодня ниже, чем вчера — emphasis first on the rate
• В банке сегодня курс ниже, чем вчера — emphasis first on in the bank

Russian has no articles.

So курс can mean:

• the rate
• a rate

Which one is meant depends on context. In this sentence, context makes it clear that it means something like the exchange rate.

This is one of the big differences from English: Russian relies on context instead of words like a and the.

Most likely, yes.

The noun курс has several meanings, including:

• rate
• exchange rate
• course
• direction

But in the context of в банке, the natural meaning is exchange rate.

So a learner should understand that Russian often relies on context to narrow down which meaning is intended.

Yes. That is exactly the kind of thing Russian allows very naturally.

The full idea could be understood as:

• ниже, чем курс был вчера
• ниже, чем вчерашний курс
• lower than the rate was yesterday

But Russian normally prefers the shorter version when the meaning is obvious:

ниже, чем вчера

This kind of omission is very common in everyday Russian.

Yes, it is completely complete and natural.

Russian often expresses ideas more compactly than English because it can omit:

• the present-tense is
• repeated nouns
• articles

So although English might want something longer, Russian is perfectly happy with:

Сегодня курс в банке ниже, чем вчера.

It is a normal, full sentence.