السباح بالمي العميقة.

Because Arabic usually leaves out the present-tense to be. In Levantine, السباح بالمي العميقة can naturally mean The swimmer is in the deep water without any separate word for is.

English needs is here, but Arabic normally does not in present-tense noun sentences.

Yes. Here السباح means the swimmer.

In fully marked spelling, it would be السَّبّاح, but everyday Arabic writing usually leaves out the short vowels and the shadda. So learners often have to know the pronunciation from context.

If you wanted the activity swimming, many speakers would use السباحة instead.

It is pronounced roughly is-sabbāḥ or es-sabbāḥ, depending on accent.

Two useful things to notice:

• س is a sun letter, so the ل in الـ is not pronounced separately.
• The ب is doubled, even though that is not shown in plain spelling here.

So you do not say al-sabaah. You say something closer to is-sabbāḥ.

It is made of:

• بـ = a preposition that often means in, at, or with in Levantine
• الـ = the
• مي = water

So بالمي means in the water.

In Levantine Arabic, بـ is very commonly used where English would use in. So بالمي is a normal colloquial way to say in the water.

A more formal or MSA-style version would more likely use في instead.

Because مي is the everyday Levantine word for water.

• مي = spoken Levantine
• ماء = Modern Standard Arabic / formal Arabic

So if you are learning spoken Levantine, مي is the natural everyday word.

Because the adjective is agreeing with المي.

In Levantine, مي is commonly treated as feminine, so the adjective is feminine too:

• المي العميقة = the deep water

If the noun is feminine, the adjective must also be feminine.

Because Arabic adjectives match the noun in definiteness as well as gender and number.

So:

• المي العميقة = the deep water
• مي عميقة = deep water

If the noun has الـ, the adjective usually has it too.

It is clearly Levantine-style, mainly because of مي for water and بـ in بالمي.

A close MSA version would be:

السَّبّاح في الماء العميق

That version sounds formal, while السباح بالمي العميقة sounds colloquial/Levantine.