Na rogu ulicy jest przejście dla pieszych, a moje kalosze stoją zawsze obok drzwi.

Because na here expresses location, so róg changes to the locative case: rogu.

• róg = corner
• na rogu = on/at the corner

Also, na rogu is the normal expression for at the street corner.

By contrast:

• w rogu usually means in the corner, for example in a room
• na róg would suggest movement toward a corner, which does not fit this sentence

So Na rogu ulicy is the natural way to say At the corner of the street.

Ulicy is the genitive singular of ulica.

In róg ulicy, Polish uses the genitive to express a relationship like the corner of the street.

So:

• ulica = street
• ulicy = of the street

This is very common in Polish:

• koniec filmu = the end of the film
• drzwi domu = the door of the house
• róg ulicy = the corner of the street

It is a fixed and very common expression meaning pedestrian crossing or crosswalk.

Literally it breaks down like this:

• przejście = passage / crossing
• dla = for
• pieszych = pedestrians

So literally: a crossing for pedestrians.

This is the standard Polish term you will see in everyday speech and in road-related language.

Because the preposition dla always takes the genitive case.

The base word is:

• pieszy = pedestrian / on foot

Its genitive plural form is:

• pieszych

So:

• piesi = pedestrians (nominative)
• dla pieszych = for pedestrians (genitive)

That is why przejście dla pieszych is correct.

Because the subject is przejście, which is singular.

• przejście = crossing
• jest = is

So:

• jest przejście = there is a crossing

If the noun were plural, you would use są:

• Na rogu ulicy są przejścia dla pieszych = There are pedestrian crossings at the street corner

In this sentence, jest is needed because it means there is.

Polish sometimes omits to be in certain kinds of sentences, especially when saying X is Y in the present tense:

• To mój brat = This is my brother

But for existence or location, jest is very often used:

• Na stole jest książka = There is a book on the table
• Na rogu ulicy jest przejście dla pieszych = There is a pedestrian crossing at the street corner

So here jest is natural and important.

Because a often links two statements while also giving a slight sense of contrast, topic shift, or meanwhile.

Here the sentence moves from one fact to another:

• Na rogu ulicy jest przejście dla pieszych, a moje kalosze stoją zawsze obok drzwi.

This is not a strong contradiction, but a feels very natural when comparing or switching focus.

Very roughly:

• i = and
• a = and / while / whereas

In many contexts, i would be grammatically possible, but a sounds more natural for this kind of two-part statement.

Because kalosze is plural, and the possessive word must agree with it.

• singular: mój kalosz = my galosh / rain boot
• plural: moje kalosze = my galoshes / rain boots

The form moje is used with plural nouns that are not masculine personal. Since kalosze refers to objects, moje is the correct form.

This is a very natural Polish usage.

Polish often uses different position verbs depending on how something is placed:

• stoi / stoją = stands / are standing
• leży / leżą = lies / are lying
• wisi / wiszą = hangs / are hanging

So shoes or boots placed upright by the door can very naturally stand in Polish:

• Buty stoją przy drzwiach = The shoes are by the door

English usually prefers just are, but Polish often chooses a more physical verb.

Yes, Polish word order is flexible, and zawsze can move.

The given version:

• moje kalosze stoją zawsze obok drzwi

is understandable and natural.

You could also say:

• moje kalosze zawsze stoją obok drzwi

This is probably the most neutral version in everyday speech.

You may also hear other placements for emphasis, but they sound more marked:

• zawsze moje kalosze stoją obok drzwi
• obok drzwi moje kalosze stoją zawsze

So the adverb position is flexible, but not every placement sounds equally neutral.

Obok means next to / beside, and it takes the genitive case.

So after obok, the following noun must be in the genitive:

• obok domu = next to the house
• obok sklepu = next to the shop
• obok drzwi = next to the door

That is why the sentence uses drzwi after obok.

Yes. Drzwi is one of those Polish nouns that is normally used only in the plural.

So even though English says door in the singular, Polish says drzwi, which is grammatically plural.

Examples:

• Te drzwi są otwarte = This door is open
• Podchodzę do drzwi = I am approaching the door
• Obok drzwi = next to the door

Also, in this sentence drzwi is in the genitive, because of obok, but its form happens to stay drzwi.

Polish normally has no articles, so it does not use separate words for a or the.

That means:

• ulica can mean a street or the street
• drzwi can mean a door or the door
• przejście can mean a crossing or the crossing

The exact meaning comes from context.

So in this sentence, English needs articles, but Polish does not:

• Na rogu ulicy jest przejście dla pieszych
  = There is a pedestrian crossing at the corner of the street

• moje kalosze stoją zawsze obok drzwi
  = my galoshes always stand next to the door

The listener understands whether something is specific or nonspecific from the situation.