У этой старой книги потрёпанный корешок, но переплёт всё ещё крепкий.

Russian often expresses possession with the pattern у + genitive, literally at/by someone or something.

So:

• У этой старой книги потрёпанный корешок
  literally: This old book has a worn spine
• more word-for-word: At this old book, [there is] a worn spine

This is a very normal Russian way to say that something belongs to something else or is one of its features.

Compare:

• У книги есть название. = The book has a title.
• У дома большая крыша. = The house has a big roof.

So у этой старой книги is not the grammatical subject. It sets up the possessor: of this old book / this old book has.

Because у requires the genitive case.

The basic dictionary forms are:

• эта = this (feminine nominative)
• старая = old (feminine nominative)
• книга = book (feminine nominative)

After у, they become genitive singular feminine:

• этой
• старой
• книги

So:

• эта старая книга = this old book
• у этой старой книги = of this old book / this old book has

This is a very useful pattern to recognize: after many prepositions, Russian changes the form of the word, and the adjective/demonstrative has to match it.

In the present tense, Russian often omits есть when it simply means is/are.

So:

• У этой старой книги потрёпанный корешок
  means This old book has a worn spine
• literally, the hidden structure is something like У этой старой книги [есть] потрёпанный корешок

In the second clause:

• переплёт всё ещё крепкий
• literally: the binding [is] still sturdy

Again, Russian usually leaves out is in the present tense.

English needs is/has, but Russian often does not.

These are related book terms, but they are not the same thing.

• корешок = the spine of a book, the narrow outer edge where the pages are attached
• переплёт = the binding, meaning the whole bound cover structure of the book

So the sentence contrasts two parts:

• the spine is worn
• but the binding is still sturdy

A helpful way to remember it:

• корешок = one specific visible part
• переплёт = the overall binding structure

Потрёпанный is an adjective meaning worn, frayed, shabby-looking from use.

It has to agree with корешок, which is:

• masculine
• singular
• nominative

So the adjective is also:

• masculine
• singular
• nominative

That gives:

• потрёпанный корешок

Other forms would appear if the noun changed:

• потрёпанная книга = a worn book
• потрёпанные страницы = worn pages

So the ending -ый / -ий / -ой on an adjective changes to match the noun it describes.

It suggests something that has been used a lot and now looks a bit damaged, worn, or shabby.

For a book spine, потрёпанный can imply things like:

• rubbed edges
• fraying
• visible wear
• a slightly battered appearance

It does not necessarily mean completely broken. It often suggests well-used rather than destroyed.

So the sentence paints a nice contrast:

• the spine looks worn
• but the binding is still structurally solid

Both can mean something like strong/sturdy, but they are used a bit differently.

Here, крепкий is the full adjective form, and it is very natural in simple description:

• Переплёт крепкий. = The binding is sturdy.

Крепок is the short form. Short forms are often more bookish, formal, or focused on the state/quality being asserted.

• Переплёт крепок is possible, but sounds less neutral and less conversational.

For everyday description, learners will very often see and use the full form:

• дом крепкий
• чай крепкий
• переплёт крепкий

So in this sentence, крепкий is the most natural neutral choice.

Всё ещё means still.

So:

• переплёт всё ещё крепкий = the binding is still sturdy

It suggests that despite the wear on the spine, the binding has remained strong.

Compare:

• Он всё ещё дома. = He is still at home.
• Она всё ещё работает. = She is still working.
• Книга всё ещё полезна. = The book is still useful.

In this sentence, всё ещё adds the idea of contrast and persistence: even though one part is worn, the binding continues to be strong.

Yes, Russian word order is flexible.

This version:

• У этой старой книги потрёпанный корешок, но переплёт всё ещё крепкий

sounds natural and descriptive. It first introduces the book, then mentions one feature, then contrasts it with another.

You could also rearrange parts for emphasis, for example:

• У этой старой книги корешок потрёпанный, но переплёт всё ещё крепкий.
• Переплёт у этой старой книги всё ещё крепкий, но корешок потрёпанный.

These are possible, but the focus changes slightly.

A rough guide:

• earlier in the sentence = often background/topic
• later in the sentence = often new or emphasized information

So the given word order feels smooth and natural for neutral description.

Because но means but, and it joins two clauses.

The two parts are:

• У этой старой книги потрёпанный корешок
• переплёт всё ещё крепкий

Russian normally uses a comma before coordinating conjunctions like но when they connect clauses.

So this is standard punctuation, just like English usually writes:

• The spine is worn, but the binding is still sturdy.

Yes, historically it is related.

• корень = root
• корешок is a diminutive form that literally means something like little root

In modern usage, though, корешок has several established meanings, including:

• the spine of a book
• a small root/rootlet in other contexts

So in book vocabulary, learners should simply remember:

• корешок книги = book spine

Even if the original imagery is interesting, the modern meaning here is straightforward and common.