Фермерський ринок працює.

Because фермерський has to agree with ринок.

• ринок = market
• It is a masculine singular noun in the nominative case
• So the adjective must also be masculine singular nominative

That gives:

• фермерський ринок = farmers’ market / farmer’s market

Compare:

• фермерський = masculine
• фермерська = feminine
• фермерське = neuter

So the ending -ий here matches ринок.

It is in the nominative case because it is the subject of the sentence.

In Фермерський ринок працює, the market is the thing doing the action:

• ринок = the subject
• працює = works / is operating / is open

So nominative is the expected case.

Ukrainian has no articles.

English distinguishes:

• a farmers’ market
• the farmers’ market

Ukrainian usually just says:

• фермерський ринок

The exact meaning depends on context. This phrase can mean:

• a farmers’ market
• the farmers’ market

You figure that out from the situation, not from a separate article word.

Literally, працює means works or is working.

But with places, businesses, services, or institutions, Ukrainian often uses працювати where English would often say:

• is open
• is operating
• is running

So in this sentence, Фермерський ринок працює is naturally understood as something like:

• The farmers’ market is open
• The farmers’ market is operating

not just The farmers’ market works in a mechanical sense.

The dictionary form is працювати = to work.

працює is:

• 3rd person singular
• present tense

So it means:

• he/she/it works
• is working
• or, with places/businesses, is open / is operating

A few forms of працювати:

• я працюю = I work
• ти працюєш = you work
• він / вона / воно працює = he / she / it works
• ми працюємо = we work
• ви працюєте = you work
• вони працюють = they work

No. The noun ринок = market is singular, so the verb is singular too.

In English, farmers’ market contains the idea of plural farmers, but the main noun is still market, which is singular. Ukrainian expresses this with the adjective фермерський, not with a plural noun.

So:

• фермерський ринок = singular
• працює = singular verb

If you wanted plural, you would say:

• фермерські ринки працюють = farmers’ markets are operating

A rough pronunciation guide:

• Фермерський ≈ fer-MER-sʹkyi
• ринок ≈ RY-nok
• працює ≈ pra-TSYU-ye

The main stresses are:

• ферме́рський
• ри́нок
• працю́є

A few pronunciation notes:

• и is not exactly English ee or ih; it is a central vowel, somewhat like a short relaxed i
• ц = ts
• ю often sounds like yu
• In працює, the sequence цю is pronounced together smoothly: tsyu

Yes, Ukrainian word order is more flexible than English, but the basic order here is the most neutral:

• Фермерський ринок працює.

This is a plain, natural statement.

Other word orders are possible for emphasis, for example:

• Працює фермерський ринок.

This can sound more like The farmers’ market is open/operating with extra focus on the fact that it is functioning.

Still, for learners, the original order is the safest and most standard.

Yes, it is a normal and common way to say it.

The adjective фермерський comes from фермер = farmer and means something like:

• farmer-related
• farm
• farmers’

So фермерський ринок is the standard phrase for farmers’ market.

Depending on context, people might also use related expressions, but фермерський ринок is the straightforward and idiomatic term.

Yes, in theory працює can mean works / functions in a general sense.

For example:

• Цей метод працює. = This method works.
• Комп’ютер не працює. = The computer isn’t working.

But with a place like ринок, the most natural meaning is usually:

• is open
• is operating

So context strongly pushes the meaning toward The farmers’ market is open / running.

It can, but it is often less natural or simply less direct in this context.

Ukrainian commonly uses працює for shops, markets, offices, services, etc.:

• Магазин працює. = The store is open.
• Банк працює. = The bank is open.
• Ринок працює. = The market is open.

You can also say things with відкритий / відчинений in some contexts, but працює is very common when talking about business hours or whether a place is operating.