Proszę o drugi ręcznik, ten jest mokry.

Yes — proszę is one of those very common Polish words with several uses.

In this sentence, proszę comes from the verb prosić and means something like I ask / I request. So Proszę o drugi ręcznik is literally I’m asking for another towel.

But in other contexts, proszę can also mean:

• please
• here you are
• go ahead
• come in

So the exact meaning depends on the situation.

Because Polish usually leaves out subject pronouns when they are already clear from the verb form.

Here, proszę already tells you it is I:

• proszę = I ask / I request

So Ja proszę o drugi ręcznik is possible, but it would usually sound more emphatic, as if you were stressing I.

Because prosić o is a fixed pattern in Polish meaning to ask for.

So:

• prosić o coś = to ask for something

That means:

• Proszę o ręcznik = I’m asking for a towel

This is something learners usually just need to memorize as a verb pattern:

• prosić o + accusative

Because prosić o takes the accusative case, and ręcznik is a masculine inanimate noun.

For masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: drugi ręcznik
• accusative: drugi ręcznik

That is why nothing visibly changes here.

For comparison, with a masculine animate noun, you would usually see a different form:

• Widzę nowego gościa = I see a new guest

It can mean both, depending on context.

In this sentence, it most naturally means another or a replacement one:

• Proszę o drugi ręcznik = I’d like another towel

But literally, drugi is also the normal word for second:

• drugi dzień = the second day

In everyday requests, Polish often uses drugi where English would naturally say another.

Yes, you could, but the nuance is a little different.

• drugi ręcznik = another towel / a second towel
• inny ręcznik = a different towel

In a hotel or similar situation, drugi ręcznik sounds very natural if you want a replacement.
Inny ręcznik is also possible, but it focuses more on different, not just one more / another.

Because Polish, like English, can avoid repeating the noun when it is already obvious.

So:

• ten jest mokry = this one is wet

The noun ręcznik is understood from context.

You could say the full version:

• ten ręcznik jest mokry

But that sounds less natural here because the noun has just been mentioned.

Because ręcznik is a masculine noun, and the demonstrative has to agree with it.

Forms of this in singular:

• ten = masculine
• ta = feminine
• to = neuter

So:

• ten ręcznik = this towel
• ta książka = this book
• to okno = this window

Since ręcznik is masculine, ten is the correct form.

Because adjectives in Polish agree with the noun in gender, number, and case.

Here, the understood noun is ręcznik, which is:

• masculine
• singular

So the adjective must also be masculine singular:

• mokry = wet (masculine singular)

Compare:

• mokry ręcznik = wet towel
• mokra koszulka = wet T-shirt
• mokre okno = wet window

Yes. It separates two closely connected clauses:

• Proszę o drugi ręcznik
• ten jest mokry

This is a normal way to write it. You could also split it into two sentences:

• Proszę o drugi ręcznik. Ten jest mokry.

That may even feel slightly clearer for a learner, but the version with the comma is perfectly natural.

Yes. Polish word order is more flexible than English word order.

The sentence you have is a very neutral, natural version:

• Proszę o drugi ręcznik, ten jest mokry.

Other versions are possible, for example:

• Ten jest mokry, proszę o drugi ręcznik.
• O drugi ręcznik proszę, ten jest mokry.

But these may sound more marked or put emphasis in a different place. For learners, the original order is a very good one to use.

Yes, it is polite and perfectly acceptable.

If you want to sound a bit softer or more formal, you could also say:

• Czy mogę prosić o drugi ręcznik? Ten jest mokry.
• Czy mógłbym / mogłabym prosić o drugi ręcznik? Ten jest mokry.

A very common service-style alternative is:

• Poproszę drugi ręcznik, ten jest mokry.

So the original sentence is good, but there are slightly softer or more conversational alternatives too.

A rough English-style guide:

• Proszę ≈ PRO-sheh
• o ≈ oh
• drugi ≈ DROO-ghee
• ręcznik ≈ RENCH-neek
• ten ≈ ten
• jest ≈ yest
• mokry ≈ MO-kri

A couple of sounds to notice:

• sz sounds like sh
• cz sounds like ch
• j sounds like English y
• ę is a nasal vowel, so rę- is not exactly the same as plain re-, though learners often start with something close to ren- or rench- as an approximation.