Na półce leżą ścierka i mop.

Because after na when you mean location in the sense of on or at, Polish often uses the locative case.

The base noun is półka = shelf.
Its locative singular form is półce.

So:

• na półce = on the shelf
• półka is just the basic dictionary form

A small spelling change happens too: k changes to c before the ending -e, which is normal in Polish declension.

This is a very important Polish pattern:

• na półce = on the shelf already there, static location
• na półkę = onto the shelf, movement toward it

So:

• Na półce leżą ścierka i mop. = They are lying on the shelf.
• Kładę ścierkę na półkę. = I put the cloth onto the shelf.

In short:

• location → often locative
• motion toward → often accusative

Leżą is the 3rd person plural present tense form of the verb leżeć, which means to lie.

A quick mini-paradigm:

• leżę = I lie
• leżysz = you lie
• leży = he/she/it lies
• leżymy = we lie
• leżycie = you all lie
• leżą = they lie

So in this sentence, leżą means are lying.

Because together they make a compound subject: ścierka i mop = the cloth and the mop.

Even though each noun is singular on its own, the whole subject is plural, so the verb must also be plural:

• ścierka leży = the cloth is lying
• mop leży = the mop is lying
• ścierka i mop leżą = the cloth and the mop are lying

Notice that the nouns themselves stay singular because there is one cloth and one mop, not several of each.

Polish often prefers more specific position verbs where English might simply use is/are.

So for objects, Polish commonly distinguishes:

• leżeć = to lie
• stać = to stand
• wisieć = to hang

Since these objects are understood as lying on the shelf, leżą sounds natural.

You could sometimes hear są, and it would be understandable, but leżą gives a more natural physical picture.

Compare:

• Na półce leżą książki. = Books are lying on the shelf.
• Na stole stoi kubek. = A mug is standing on the table.
• Na ścianie wisi obraz. = A picture is hanging on the wall.

They are in the nominative case, because they are the subject of the sentence.

You can tell because they are the things doing the lying:

• ścierka = cloth
• mop = mop

Both are in their basic dictionary forms here.

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

This sentence could also be:

• Ścierka i mop leżą na półce.

That version is also correct.

The difference is mostly about focus or information structure:

• Na półce leżą ścierka i mop.
  Emphasizes where they are
• Ścierka i mop leżą na półce.
  Emphasizes what is on the shelf

Both mean the same thing overall.

Because Polish does not have articles like English a/an and the.

So półka can mean:

• a shelf
• the shelf

and the exact meaning depends on context.

That is completely normal in Polish. Learners often want to add a separate word for the, but Polish simply does not work that way.

A rough guide:

• na = like nah
• półce ≈ POO-wt-seh
  
  • ó sounds like oo
  • ł sounds like English w
  • c sounds like ts
• leżą ≈ LEH-zhaw̃
  
  • ż sounds like the s in measure
  • ą is a nasal vowel
• ścierka ≈ SHCHER-kah or SH-cher-kah
  
  • ś is a soft sh
  • ci before a vowel makes a soft ch/sh-like sound
• mop = very close to English mop

Polish stress is usually on the second-to-last syllable:

• PÓŁ-ce
• LE-żą
• ŚCIER-ka

Yes. I is the standard Polish word for and.

So:

• ścierka i mop = the cloth and the mop

Also, just like in English, you normally do not put a comma before i when it simply joins two nouns.