kono mise no keeki ha ninki ga takakute, mettani tokubai ni naranai.

この店のケーキ means the cakes of this shop / this shop’s cakes.

• この = this
• 店 = shop/store
• の = links nouns, similar to of or an apostrophe-s
• ケーキ = cake / cakes

So この店のケーキ means the cakes sold by this shop.

If you said このケーキ, that would mean this cake, referring to one particular cake near the speaker.

The は marks この店のケーキ as the topic of the sentence.

So the sentence is saying, roughly:

• As for this shop’s cakes, their popularity is high, and they rarely go on sale.

This is very common in Japanese:

• A は B が ...
• As for A, B is ...

Here:

• この店のケーキ は = as for this shop’s cakes
• 人気 が 高くて = their popularity is high

Both 人気が高い and 人気がある are natural and mean something like to be popular.

The difference is mostly nuance:

• 人気がある = there is popularity; a very common, straightforward way to say is popular
• 人気が高い = the level of popularity is high; slightly more descriptive or evaluative

In this sentence, 人気が高い emphasizes that the cakes are very popular / highly popular.

In 人気が高い, 人気 is the thing being described as high.

So:

• 人気 = popularity
• が marks it as the subject of the adjective
• 高い = high

Literally, it is like saying:

• popularity is high

This is why Japanese often uses patterns like:

• 値段が高い = the price is high
• 身長が高い = the height is high / someone is tall
• 人気が高い = popularity is high

高くて is the -て form of 高い.

Here it connects two related ideas:

• 人気が高くて
• めったに特売にならない

So it means something like:

• because they are popular, they rarely go on sale or
• they are popular, and as a result they rarely go on sale

The -て form often connects clauses and can imply:

• simple sequence
• two related descriptions
• cause/reason

In this sentence, a reason-like connection is very natural.

Yes, 人気で、めったに特売にならない is also possible.

That version treats 人気 more directly as the reason:

• because they’re popular, they rarely go on sale

The original 人気が高くて is a bit fuller and more descriptive:

• their popularity is high, so...

Both are natural, but the original highlights the degree of popularity more clearly.

めったに almost always goes with a negative form and means:

• rarely
• seldom
• hardly ever

So:

• めったに～ない = rarely do ~

Examples:

• めったに行かない = I rarely go
• めったに見ない = I rarely see
• めったに特売にならない = it rarely goes on sale

Using めったに without a negative verb is generally incorrect in standard Japanese.

特売 means special sale, sale price, or special bargain sale.

～になる means to become ~.

So:

• 特売になる = to become a sale item, to go on special, to be put on sale

In this sentence:

• めったに特売にならない = they rarely go on sale

It does not mean the cakes become cheap by themselves; it means the store rarely sells them at a special discount price.

The particle に is often used with なる to mark the result of a change.

Pattern:

• A に なる = become A

Examples:

• 先生になる = become a teacher
• 春になる = become spring / it becomes spring
• 有名になる = become famous
• 特売になる = become a sale item / go on sale

So 特売に is the state/result, and ならない means does not become.

Because the verb is なる (to become), not a simple adjective or noun sentence.

The structure is:

• 特売になる = to go on sale
• negative: 特売にならない = does not go on sale

You cannot say 特売ない here because 特売 is a noun, and the grammar needed is noun + に + なる.

It can be understood as cake or cakes, depending on context.

Japanese nouns usually do not show singular/plural the way English nouns do. So:

• この店のケーキ can mean
  • this shop’s cake
  • this shop’s cakes

In this sentence, English will often translate it as this shop’s cakes because the statement is general:

• they are popular
• they rarely go on sale

Grammatically, it is closer to:

• the popularity of this shop’s cakes is high

But in natural English, we usually say:

• this shop’s cakes are very popular

Japanese often expresses things through nouns like 人気 rather than always using a direct adjective equivalent.

So the literal structure and the natural translation are slightly different.

It breaks down like this:

• この店のケーキは = as for this shop’s cakes
• 人気が高くて、 = their popularity is high / they are very popular, and so...
• めったに = rarely
• 特売に = into a sale item / on special
• ならない = do not become

So the whole sentence is built as:

• Topic + description/reason + negative result

A very literal gloss would be:

• As for this shop’s cakes, their popularity is high, so they rarely become sale items.

Natural English:

• This shop’s cakes are very popular, so they rarely go on sale.

Yes. The sentence suggests a reason-and-result relationship:

• the cakes are popular
• because demand is high, the store usually does not need to discount them

So it implies:

• they sell well anyway
• the shop has little reason to lower the price

That nuance comes from 人気が高くて connecting naturally to めったに特売にならない.