Этот стул крепкий, но старый.

In Russian, the verb to be is usually left out in the present tense.

So:

• Этот стул крепкий, но старый. = This chair is sturdy, but old.

Russian does not normally say a present-tense equivalent of is here.
If you want past or future, then forms of быть do appear:

• Этот стул был крепкий, но старый. = This chair was sturdy, but old.
• Этот стул будет крепкий. = This chair will be sturdy.

Because стул is a masculine singular noun, and the demonstrative this must agree with it.

Russian this changes by gender:

• этот = masculine
• эта = feminine
• это = neuter

Since стул is masculine, you say:

• этот стул

Compare:

• эта книга = this book
• это окно = this window

Because adjectives in Russian must agree with the noun they describe in:

• gender
• number
• case

Here, стул is:

• masculine
• singular
• nominative

So the adjectives also appear in masculine singular nominative form:

• крепкий
• старый

If the noun changed, the adjectives would change too. For example:

• эта книга старая = this book is old
• это кресло старое = this armchair is old
• эти стулья старые = these chairs are old

Here крепкий means something like sturdy, solid, or strongly built.

For furniture, крепкий suggests that the chair is:

• not fragile
• well-made
• able to hold up well

So in this sentence, крепкий is more like sturdy than just general strong.

Russian крепкий can also be used in other contexts, for example:

• крепкий чай = strong tea
• крепкий человек = a strong/tough person
• крепкий дом = a solid house

Because но means but, and in Russian it normally introduces a new clause or a contrasting part of the sentence, so it is preceded by a comma.

So:

• Этот стул крепкий, но старый.

This is similar to English, where we often also put a comma before but in a sentence like:

• This chair is sturdy, but old.

Both но and а can connect contrasting ideas, but they are not identical.

• но = but, often a stronger contradiction
• а = often softer contrast, sometimes closer to while, whereas, or and

In this sentence, но is very natural because the idea is:

• the chair is sturdy, but it is old

That feels like a clear contrast.

You might also hear:

• Этот стул крепкий, а старый.

But that sounds unnatural here.
A more natural use of а would be in comparisons such as:

• Этот стул крепкий, а тот слабый. = This chair is sturdy, while that one is weak.

It is in the nominative case.

That is because стул is the subject of the sentence:

• Этот стул = this chair

The nominative is the basic dictionary form of the noun, and it is commonly used for the subject.

You can also see that the adjectives and этот match the nominative masculine singular form:

• этот
• крепкий
• старый

Yes. Russian word order is more flexible than English word order, because endings carry a lot of the grammar.

The neutral order here is:

• Этот стул крепкий, но старый.

But you could change the order for emphasis:

• Этот стул старый, но крепкий. This puts old first and changes the emphasis.
• Крепкий этот стул, но старый. This is more marked and sounds more stylistic or emphatic.

So yes, word order can change, but the original version is the most natural neutral phrasing.

The stress is:

• Э́тот стул кре́пкий, но ста́рый.

A rough pronunciation guide:

• Этот = EH-tuht
• стул = stool
• крепкий = KRYEP-kee
• но = no
• старый = STA-ryy

A couple of useful notes:

• ё is not used here; the first word is э́тот
• крепкий has a consonant cluster, so learners often need practice saying it smoothly

Both are normal masculine singular nominative adjective endings.

In dictionaries, many adjectives appear as:

• -ый
• -ий
• sometimes -ой

Which ending appears depends on spelling patterns and the word itself.
So:

• старый is just the normal dictionary form of that adjective
• крепкий is also the normal dictionary form of that adjective

You do not choose between -ый and -ий freely; you learn the adjective in its standard form.

The important point for a learner is that both forms can be masculine singular nominative.

Yes, absolutely.

• Этот стул крепкий, но старый. = This chair is sturdy, but old.
• Стул крепкий, но старый. = The chair is sturdy, but old / A chair is sturdy, but old, depending on context

Adding этот makes it specifically this chair.
Without этот, the sentence is less specific and depends more on context.

Everything that agrees with the noun would change:

• this
• the adjectives

Examples:

• Эта книга крепкая, но старая.
  feminine singular

• Это кресло крепкое, но старое.
  neuter singular

• Эти стулья крепкие, но старые.
  plural

This is a good example of how Russian agreement works: the demonstrative and adjectives must match the noun.