isogasii hi ha hirugohan wo tabenaikoto mo arimasu.

Because 忙しい is an い-adjective, and い-adjectives directly modify nouns.

• 忙しい日 = a busy day / busy days
• 忙しい日は = as for busy days / on busy days

This is very normal Japanese word order: modifier first, noun second.

Because 忙しい is an adjective, not a noun.

• Adjectives modify nouns directly: 忙しい日
• Nouns usually need の to modify another noun: for example, 日本の車

So 忙しいの日 would be incorrect.

Here, は marks 忙しい日 as the topic or frame of the sentence.

So 忙しい日は means something like:

• As for busy days...
• On busy days...

It can also carry a slight contrastive feeling, like at least on busy days or when it’s a busy day, this is what happens.

It can mean either, depending on context, but in this sentence it most naturally means on busy days in general.

Japanese often does not mark singular vs. plural as clearly as English does. So:

• 忙しい日 could be a busy day
• or busy days

The rest of the sentence makes it sound like a general habit or occasional pattern, so on busy days is the most natural reading.

Because 昼ご飯 is the direct object of 食べる.

• 昼ご飯を食べる = to eat lunch

The particle を marks what is being eaten.

The subject is omitted, which is very common in Japanese.

From context, it is usually understood as something like:

• I
• we
• or another contextually obvious person

So the sentence often implies:

• I sometimes don’t eat lunch on busy days.

Japanese leaves out subjects when they are already clear from context.

こと turns the verb phrase into a noun-like expression.

• 食べない = do not eat
• 食べないこと = not eating / the fact of not eating

So instead of directly saying I don’t eat lunch, the sentence talks about the event/case of not eating lunch.

This is a very common grammar pattern in Japanese.

Because the meaning is different.

• 昼ご飯を食べません = I do not eat lunch
  This sounds like a straightforward statement or habit.
• 昼ご飯を食べないこともあります = There are also times when I don’t eat lunch
  This means sometimes or occasionally.

So Vないこともある is a common pattern for saying:

• sometimes do not V
• there are times when one does not V

It sounds less absolute and more natural for occasional situations.

Here, も adds the sense of also or sometimes, among other possibilities.

Literally, the phrase is close to:

• There are also cases of not eating lunch.

In natural English, this becomes:

• Sometimes I don’t eat lunch.

So も helps give the sentence its occasional / not always nuance.

Because ある is used for things, events, and abstract matters, while いる is used for people and animals.

Here, 食べないこと is an abstract thing/event:

• the fact/case of not eating lunch

So Japanese uses ある:

• 食べないことがある
• 食べないこともあります

Also, あります is just the polite form of ある.

Yes. The plain form would be:

• 忙しい日は昼ご飯を食べないこともある。

The sentence you have uses polite style:

• 忙しい日は昼ご飯を食べないこともあります。

Both mean the same thing; the difference is mainly politeness and tone.

A very natural understanding is:

• Sometimes, on busy days, I don’t eat lunch.
• On busy days, there are times when I don’t eat lunch.

The important nuance is that this is not always true. It only happens sometimes.