У тижні сім днів.

Because у meaning in normally requires the locative case.

• Dictionary form: тиждень = week
• Locative singular: у тижні = in a week / in the week

So тижні is not a plural here. It is a singular locative form.

After numerals 5 and higher, Ukrainian nouns usually take the genitive plural.

So:

• один день
• два / три / чотири дні
• п’ять / шість / сім днів

That is why the sentence has сім днів.

In Ukrainian, the present-tense verb to be is often omitted in simple statements.

So Ukrainian can say:

• У тижні сім днів.

where English needs something like:

• There are seven days in a week.

You can also say У тижні є сім днів, with є, but it is not required.

Yes. That is grammatical.

• У тижні сім днів. = more neutral, compact
• У тижні є сім днів. = more explicit, a bit more like there are

In everyday Ukrainian, leaving out є in this kind of sentence is very common.

Yes, Ukrainian word order is fairly flexible.

This order puts the setting first:

• У тижні = the context
• сім днів = the main information

You could also hear:

• Сім днів у тижні.

The basic meaning stays the same, but the emphasis shifts slightly. The original version sounds very natural for stating a general fact.

Usually yes. У and в often alternate for sound and flow.

So both are possible in many contexts:

• у тижні
• в тижні

At the beginning of this sentence, у тижні sounds smoother to many speakers because в тижні starts with a heavier consonant cluster.

So the choice is often about euphony, not a big change in meaning.

No, not here.

• у тижні here means something like within a week / in a week as a unit
• на тижні usually means during the week or this week / sometime this week

So for this sentence, у тижні is the correct choice. На тижні сім днів would not sound right with this meaning.

A rough pronunciation is:

u TYZH-ni sim dniv

A few helpful points:

• ж sounds like the s in measure
• і sounds like ee in see
• днів begins with a real dn- cluster, with no extra vowel inserted

So the rhythm is roughly:

u TYZH-ni sim dniv

Ukrainian has no articles. There is no direct equivalent of English a/an or the.

So Ukrainian simply says:

• У тижні сім днів

and context tells you that this is a general fact, not a specific one.

That is completely normal in Ukrainian.