Na okładce było zdjęcie kobiety, a w środku znalazłam zakładkę z adresem księgarni.

Because zdjęcie is a neuter noun in Polish.

In the past tense, być changes to match the gender of the subject:

• był = masculine
• była = feminine
• było = neuter

Since zdjęcie means photo/picture and is neuter, Polish uses:

• było zdjęcie = there was a photo

Even though okładka is feminine, it is not the grammatical subject here. The subject is zdjęcie.

Because kobiety is in the genitive case after zdjęcie.

Polish often uses this pattern:

• zdjęcie kogo? czego? = a photo of whom? of what?

So:

• kobieta = woman
• zdjęcie kobiety = photo of a woman

This is a very common structure. Compare:

• portret mężczyzny = a portrait of a man
• opis miasta = a description of a city
• kolor domu = the color of the house

Because na can mean on, and here it is followed by the locative case to express location.

• okładka = cover
• na okładce = on the cover

The noun changes form:

• okładka → okładce

This is the normal pattern after na when talking about where something is.

Compare:

• na stole = on the table
• na zdjęciu = in the photo / on the photo
• na stronie = on the page

Because w means in, and w środku is a fixed expression meaning inside or in the middle/interior.

• środek = middle, interior
• w środku = inside

In this sentence, it means something like inside the book / inside it.

This phrase is very common:

• W środku było zdjęcie. = There was a photo inside.
• Co jest w środku? = What is inside?

Because znalazłam is the past tense, feminine singular form of znaleźć = to find.

Polish past tense agrees with the gender of the speaker:

• znalazłem = I found, said by a man
• znalazłam = I found, said by a woman

So this sentence tells you that the speaker is female.

Polish often leaves out ja (I) because the verb form already shows the person, and here it also shows the gender.

Because Polish often drops subject pronouns when they are not needed.

The verb already tells you the person:

• znalazłam = I found (female speaker)

So ja is unnecessary unless you want emphasis:

• Ja znalazłam zakładkę... = I found the bookmark...
  This sounds more emphatic, like contrasting with someone else.

This is very normal in Polish.

Not exactly.

• i usually means and
• a can also mean and, but often with a sense of contrast, switch of focus, or meanwhile

In this sentence:

• Na okładce było zdjęcie kobiety, a w środku...

The meaning is roughly:

• There was a photo of a woman on the cover, and inside...

But a helps move from one detail to another: on the cover... and inside...

So it is not a strong contrast like but, but it is a little more contrastive than plain i.

Zakładka most commonly means bookmark.

So:

• znalazłam zakładkę = I found a bookmark

Depending on context, zakładka can also mean other things, such as a tab in a browser or a bookmark tab, but in this sentence, because of na okładce and w środku, the book-related meaning bookmark is the most natural one.

Because the preposition z here means with, and after z in this meaning, Polish uses the instrumental case.

• adres = address
• z adresem = with an address

So:

• zakładkę z adresem księgarni = a bookmark with the bookstore’s address

Compare:

• kawa z mlekiem = coffee with milk
• kobieta z dzieckiem = a woman with a child
• list z podpisem = a letter with a signature

Because księgarni is in the genitive case after adres.

The pattern is:

• adres kogo? czego? = address of whom? of what?

So:

• księgarnia = bookstore
• adres księgarni = the bookstore’s address / address of the bookstore

This is another very common genitive structure in Polish.

Compare:

• numer telefonu = phone number
• drzwi domu = the door of the house
• nazwa miasta = the name of the city

By itself, Polish usually does not mark the vs a the way English does, because Polish has no articles.

So zdjęcie kobiety could mean:

• a photo of a woman
• the photo of the woman

The exact meaning depends on context. In this sentence, English would most naturally say a photo of a woman unless the woman was already known.

This is something English speakers often need to get used to: Polish leaves definiteness to context.

Yes. Polish word order is more flexible than English, because case endings show the grammatical relationships.

The original sentence is natural:

• Na okładce było zdjęcie kobiety, a w środku znalazłam zakładkę z adresem księgarni.

But other orders are also possible, depending on emphasis. For example:

• Zdjęcie kobiety było na okładce...
• W środku znalazłam zakładkę...
• Zakładkę z adresem księgarni znalazłam w środku.

The original version sounds very natural because it presents the locations first:

• On the cover...
• Inside...

That gives the sentence a nice descriptive flow.

Znaleźć is perfective.

That means it refers to a completed action: to find, in the sense of successfully finding something.

Its imperfective partner is usually:

• znajdować / znajdować się in some contexts, or more commonly for repeated searching/finding contexts znajdować

For example:

• Znalazłam zakładkę. = I found the bookmark.
• Często znajdowałam w książkach różne notatki. = I often used to find various notes in books.

In your sentence, the completed action is exactly what is needed: at some point, the speaker found the bookmark.

Yes, very close.

Polish often uses być in this way to introduce the existence or presence of something:

• Na stole była książka. = There was a book on the table.
• W pokoju było zimno. = It was cold in the room.
• Na okładce było zdjęcie kobiety. = There was a photo of a woman on the cover.

So although the structure is not identical to English, translating it as there was is very natural.