Pastane bakkaldan daha uzak ama kasaptan daha yakın.

Because Turkish usually marks the thing you compare against with the ablative case: -dan / -den / -tan / -ten.

So:

• bakkaldan daha uzak = farther than the grocer/store
• kasaptan daha yakın = closer than the butcher/butcher's shop

This suffix also follows normal sound rules:

• after a voiced consonant, you get -dan / -den
• after a voiceless consonant, you get -tan / -ten

So:

• bakkal + dan → bakkaldan
• kasap + tan → kasaptan

Here daha is the comparative marker, meaning more.

Turkish usually forms comparatives with:

daha + adjective/adverb

So:

• daha uzak = farther / more distant
• daha yakın = closer / nearer

Unlike English, Turkish does not usually add an -er ending to the adjective.

Because daha modifies the adjective directly.

The normal pattern is:

X, Y-dan daha adjective

So:

• bakkaldan daha uzak
• kasaptan daha yakın

Not daha bakkaldan uzak.

You can think of daha as belonging with the adjective, not with the noun.

In Turkish, the 3rd person present-tense copula is often omitted.

So a sentence like:

• Pastane ... uzak
• Pastane ... yakın

already means:

• The pastry shop is far...
• The pastry shop is near...

This is completely normal Turkish.
An explicit ending or form for is is often unnecessary in everyday speech.

Because there are two separate comparisons:

• bakkaldan daha uzak
• kasaptan daha yakın

Each adjective has its own comparative meaning, so each one normally gets its own daha.

If you leave out the second daha, it usually sounds incomplete or less natural.

Good question. This is because kasaptan is not the place the pastry shop is near to. It is the thing being compared against.

Compare:

• Okula yakın = near the school
  Here okul is the thing something is near to, so Turkish uses the dative: -a / -e

But in your sentence:

• kasaptan daha yakın

means:

• closer than the butcher's shop is
• or closer than the butcher

So this is a comparison, and Turkish uses the ablative: -dan / -den / -tan / -ten

Ama means but.

It is very common and natural in everyday Turkish.

So:

• ... daha uzak ama ... daha yakın = ... farther, but ... closer

Other possible words are:

• fakat
• ancak

But ama is probably the most common conversational choice.

Yes. This is a very natural structure.

The sentence is organized like this:

• Pastane = subject
• bakkaldan daha uzak = first comparison
• ama = contrast
• kasaptan daha yakın = second comparison

Turkish often puts the descriptive/predicate part toward the end, so this sounds normal.

A comma after Pastane is optional in writing:

• Pastane, bakkaldan daha uzak ama kasaptan daha yakın.

They can mean either, depending on context.

Very often in Turkish:

• bakkal = the grocer or the grocery shop
• kasap = the butcher or the butcher's shop

In this sentence, English will usually understand them as the grocery store and the butcher's shop, because the sentence is comparing locations.

Turkish often lets context tell you whether it means the person, the business, or the place.

The reference point is simply understood from context.

Just like English, Turkish can leave that unstated if everyone already knows what the comparison is about.

For example, the missing reference point might be:

• our house
• here
• the school
• the bus stop

So the sentence really means something like:

• The pastry shop is farther than the grocery store but closer than the butcher's shop, relative to the same understood place.

Yes. They are functioning as predicate adjectives.

• uzak = far
• yakın = near

In this sentence they describe pastane.

Turkish adjectives do not change form for number or gender, so uzak and yakın stay the same.