Ten kubek jest na stole, a ten drugi jest w kuchni.

Because ten has to agree with kubek in gender, number, and case.

• kubek is masculine singular
• so the correct demonstrative is ten
• ta is for feminine nouns
• to is for neuter nouns, or can function as this/that as a pronoun in other structures

So:

• ten kubek = this mug/cup
• ta filiżanka = this cup if the noun is feminine
• to okno = this window if the noun is neuter

Kubek usually means a mug or a cup-like container, often something with a handle. In everyday Polish, it often feels closer to mug than to a delicate tea cup.

Some related words:

• kubek = mug / cup
• filiżanka = teacup / coffee cup
• szklanka = glass / drinking glass

So in this sentence, kubek is a normal everyday drinking mug or cup.

Because na stole expresses location: the mug is on the table.

After na, Polish often uses:

• locative for location
• accusative for movement toward a place

So:

• Ten kubek jest na stole. = The mug is on the table.
  
  • location
  • stole = locative of stół

But:

• Połóż kubek na stół. = Put the mug onto the table.
  
  • movement toward the table
  • stół here is in the accusative, which happens to look the same as the nominative for this noun

This location vs. movement contrast is very important in Polish prepositions.

Because na here requires the locative case when it means on in a location sense.

The noun changes like this:

• stół = table
• na stole = on the table

So stole is simply the locative singular form of stół.

This is one of the normal noun changes learners have to get used to in Polish: nouns often change their endings after prepositions.

For the same basic reason: w meaning in takes the locative when you are talking about location.

• kuchnia = kitchen
• w kuchni = in the kitchen

So kuchni is the locative singular form of kuchnia.

Compare:

• jest w kuchni = is in the kitchen
• idzie do kuchni = goes to the kitchen

Notice that Polish uses different prepositions and cases depending on whether you mean being somewhere or going somewhere.

Because the meanings are different:

• na usually means on a surface
• w usually means in an enclosed space or inside something

So:

• na stole = on the table
• w kuchni = in the kitchen

This matches English pretty well in this sentence.

That said, Polish prepositions are not always identical to English ones. Some places are just learned as set expressions, so it is good to memorize them together with the noun.

Here a links two clauses with a slight sense of contrast or comparison.

So:

• Ten kubek jest na stole, a ten drugi jest w kuchni.

means something like:

• This mug is on the table, and the other one is in the kitchen
• or This mug is on the table, while the other one is in the kitchen

By contrast, i is a more neutral and, just adding information.

Very roughly:

• i = and
• a = and / while / whereas / but, depending on context

In this sentence, a is natural because two different mugs are being contrasted by location.

Ten drugi means the other one or that second one, depending on context.

Here:

• ten = this/that
• drugi = second / other

In this sentence, ten drugi refers back to another kubek without repeating the noun.

So the full idea is:

• ten drugi [kubek] = the other mug

Polish often omits the noun if it is already clear from context.

Because drugi often means the second or the other one, especially when there are two items in mind.

So if you have two mugs:

• one is on the table
• the other one is in the kitchen

then drugi is very natural.

Inny means different or another, but it does not always have that neat the second of the two feeling.

So:

• ten drugi = that other one / the second one
• ten inny = that different one / another one

In this sentence, drugi is the better fit.

Yes, a drugi jest w kuchni is also possible in many contexts.

But ten drugi is more explicit and often feels like that other one or the second one. It clearly points back to the previously understood noun.

So:

• a drugi jest w kuchni = and the other/second one is in the kitchen
• a ten drugi jest w kuchni = and that other one is in the kitchen

The version with ten sounds a bit more definite and pointing.

Because each half is its own clause, and each clause needs its verb.

• Ten kubek jest na stole
• a ten drugi jest w kuchni

Polish does sometimes omit words that are understood from context, but in a normal sentence like this, keeping jest in both clauses is standard and natural.

Since the sentence is about location and means is on / is in, the verb jest is important here.

Yes. Polish word order is fairly flexible, although the version given is the most neutral and straightforward.

The original sentence:

• Ten kubek jest na stole, a ten drugi jest w kuchni.

is a normal default order.

But Polish could move things around for emphasis, for example:

• Na stole jest ten kubek, a w kuchni ten drugi.
• Ten drugi jest w kuchni, a ten kubek jest na stole.

These versions shift the focus a little, but the basic meaning stays the same.

For a learner, the original word order is the safest one to use first.