sarada no tame ni, kyuuri wo tiisaku kirimasu.

Because サラダ is a noun, and nouns usually connect to ために with の.

• noun + のために → for, for the purpose of
• verb (dictionary form) + ために → in order to

So:

• サラダのために = for the salad
• サラダを作るために = in order to make a salad

The の is just the normal way to link the noun サラダ to ために.

Here, ために shows purpose.

So サラダのために means something like:

• for the salad
• for use in the salad
• for making a salad

It does not mean the salad is benefiting like a person would. It simply tells you the reason or purpose for cutting the cucumber.

を marks the direct object of the verb.

The verb is 切ります = cut.
The thing being cut is きゅうり.

So:

• きゅうりを切ります = cut the cucumber

In other words, を tells you what receives the action of cutting.

Because 小さい is an い-adjective, and when an い-adjective modifies a verb, it changes to the -く form.

• 小さい = small
• 小さく = small-ly / into small pieces

So:

• 小さいきゅうり = a small cucumber
• きゅうりを小さく切ります = cut the cucumber small / cut the cucumber into small pieces

Here 小さく describes how the cucumber is cut.

Yes, very literally it does. But in natural English, it usually means:

• cut into small pieces
• cut finely
• cut it small

Japanese often uses an adjective in adverb form like this:

• 細かく切る = cut finely
• 薄く切る = cut thinly
• 小さく切る = cut small / into small pieces

So this is a very common pattern.

Because Japanese normally puts the main verb at the end of the sentence.

A basic Japanese word order is often:

• topic/time/place/etc.
• object
• verb

So this sentence is structured like:

• サラダのために = for the salad
• きゅうりを = the cucumber
• 小さく切ります = cut into small pieces

This is normal Japanese sentence order.

The subject is omitted because Japanese often leaves out information that is obvious from context.

Depending on the situation, it could mean:

• I cut the cucumber small for the salad
• We cut the cucumber small for the salad
• Cut the cucumber small for the salad in recipe-style language

In Japanese, if the context already makes the subject clear, it is usually not stated.

切ります is the polite non-past form.

That means it can refer to:

• a future action: will cut
• a habitual action: cut(s)
• instructions or recipe steps: cut

In a cooking context, it often sounds like a recipe instruction or procedural statement:

• きゅうりを小さく切ります。 = Cut the cucumber into small pieces.

So even though it is not an English-style imperative, it works naturally in recipe directions.

Yes. 切ります is the polite ます-form of 切る.

Compare:

• 切る = plain form
• 切ります = polite form

In textbooks, explanations, and recipe-style writing, ます-form is very common because it sounds neat and polite.

No. Normal Japanese writing usually does not use spaces between words.

This sentence would normally be written as:

サラダのために、きゅうりを小さく切ります。

The spaces in your example are probably there just to help learners see the parts more clearly.

Grammatically, yes, ために can sometimes mean for the sake of. But in this sentence, that is not the natural interpretation.

Because this is a cooking context, サラダのために is understood as:

• for the salad
• for use in the salad
• in preparation for the salad

So context makes the meaning clear.