حدة البصر وفحص النظر مهمان.

Because this is a present-tense nominal sentence in Arabic. In Modern Standard Arabic, the verb to be is usually not stated in the present tense.

So:

حِدَّةُ البَصَرِ وَفَحْصُ النَّظَرِ مُهِمَّانِ

literally works like:

Visual acuity and eye examination important

but in natural English we translate it as:

Visual acuity and an eye exam are important.

If you wanted the past or future, Arabic would use a verb such as كان.

It is a nominal sentence with:

• a compound subject: حِدَّةُ البَصَرِ وَفَحْصُ النَّظَرِ
• a predicate: مُهِمَّانِ

So the structure is:

• حِدَّةُ البَصَرِ = visual acuity / sharpness of vision
• وَ = and
• فَحْصُ النَّظَرِ = eye exam / vision test
• مُهِمَّانِ = important, with dual agreement

A close grammatical gloss would be:

[Visual acuity] and [eye examination] [are important].

Because it is dual.

The sentence has two things joined by و:

• حِدَّةُ البَصَرِ
• فَحْصُ النَّظَرِ

Since there are exactly two items, the predicate agrees in the dual, so we get:

• مُهِمّ = important
• مُهِمَّانِ = two things are important

This is very similar to examples like:

• زَيْدٌ وَعَمْرٌو قائِمانِ = Zayd and ʿAmr are standing

Because the two nouns are mixed in gender:

• حِدَّة is feminine
• فَحْص is masculine

When Arabic has a mixed-gender pair, agreement normally defaults to the masculine. So the predicate becomes masculine dual:

• مُهِمَّانِ = masculine dual

If both nouns were feminine, then مهمتان would make sense.

Because it is the predicate of the sentence, and predicates in Arabic are often indefinite.

So even though the subject is definite, the predicate can still be indefinite:

• الولدُ طويلٌ = The boy is tall
• الكتابُ جديدٌ = The book is new
• حِدَّةُ البَصَرِ وَفَحْصُ النَّظَرِ مُهِمَّانِ = Visual acuity and eye examination are important

If you added الـ and said المهمان, that would usually change the meaning and sound unnatural here.

Because they are the first words in an iḍāfa construction.

Both of these are possessive/genitive phrases:

• حِدَّةُ البَصَرِ = the sharpness of vision
• فَحْصُ النَّظَرِ = the examination of sight / eye exam

In an iḍāfa, the first noun usually:

• does not take الـ
• does not take tanwīn

The second noun determines definiteness. Since البصر and النظر are definite, the whole phrases are definite.

So:

• حِدَّةُ البَصَرِ is definite
• فَحْصُ النَّظَرِ is definite

even though حِدَّة and فَحْص themselves do not have الـ.

Fully vowelled, the sentence is:

حِدَّةُ البَصَرِ وَفَحْصُ النَّظَرِ مُهِمَّانِ

Here is why:

• حِدَّةُ: nominative, because it is part of the subject
• البَصَرِ: genitive, because it is the second term of the iḍāfa
• فَحْصُ: nominative, also part of the subject
• النَّظَرِ: genitive, second term of the iḍāfa
• مُهِمَّانِ: nominative dual, because it is the predicate

So the important endings are:

• ـُ on حِدَّةُ and فَحْصُ
• ـِ on البَصَرِ and النَّظَرِ
• ـانِ on مُهِمَّانِ

النظر is pronounced an-naẓar, not al-naẓar.

That is because ن is a sun letter, so the ل of الـ is assimilated:

• النظر → an-naẓar

But البصر is different. ب is a moon letter, so the ل is pronounced:

• البصر → al-baṣar

So a careful pronunciation of the whole sentence is approximately:

ḥiddatu l-baṣari wa-faḥṣu n-naẓari muhimmāni

A simpler learner-friendly version is:

ḥiddat al-baṣar wa-faḥṣ an-naẓar muhimmān

Because و is a clitic in Arabic. It is written attached to the following word, not as a separate word.

So:

• وَفَحْصُ = and examination

This is normal Arabic spelling, just like:

• وَالْكِتابُ = and the book
• وَجاءَ = and he came

Not exactly, though they overlap in meaning.

• البصر usually refers to vision, eyesight, the ability to see
• النظر can refer to looking, sight, visual function, and in medical or everyday usage it appears in the common expression فحص النظر = eye exam / vision test

So in this sentence:

• حِدَّةُ البَصَرِ is something like visual acuity
• فَحْصُ النَّظَرِ is an eye examination / vision test

They are related, but not interchangeable word-for-word here.

Here it is best understood as a noun phrase, not as a full verbal phrase.

• فَحْص is a noun meaning examination
• النَّظَر is the second noun in the iḍāfa

So:

فَحْصُ النَّظَرِ = the examination of vision = eye exam / vision test

If Arabic wanted a full verbal idea such as examining vision, it would usually use an actual verb form in context. Here, the structure is clearly nominal.