Пусть гранат и вкусный, он быстро пачкает пальцы.

In this sentence, пусть does not mean let in the literal command sense. Here it is part of a concessive pattern:

пусть ... и ... = although ..., even if ..., granted that ...

So:

Пусть гранат и вкусный...
means something like:
Although pomegranate is tasty... / Granted, pomegranate is tasty...

This use of пусть is fairly common in more formal or literary-style Russian.

The и helps form the concessive expression with пусть.

So the pattern is:

• пусть X и Y, ...
• although X is Y, ...

In your sentence:

• пусть гранат и вкусный
• literally: although pomegranate is tasty

This и does not simply mean and here. It is part of the structure that gives the meaning even though / although.

A very similar idea appears in other sentences like:

• Пусть он и молод, но очень опытен.
  Although he is young, he is very experienced.

Russian usually omits the verb to be in the present tense.

So:

• гранат вкусный literally looks like
• pomegranate tasty

but it means

• pomegranate is tasty

This is completely normal in Russian.

Compare:

• Он умный. = He is smart.
• Суп горячий. = The soup is hot.

So there is no missing word here from a Russian point of view.

Вкусный agrees with гранат in gender, number, and case.

Гранат is:

• masculine
• singular
• nominative

So the adjective must also be:

• masculine
• singular
• nominative

That gives вкусный.

If the noun changed, the adjective would change too:

• груша вкусная = the pear is tasty
• яблоко вкусное = the apple is tasty
• фрукты вкусные = the fruits are tasty

Because it is the subject of the clause.

In Пусть гранат и вкусный, the thing being described as tasty is гранат, so it stays in the nominative:

• гранат = subject
• вкусный = predicate adjective agreeing with the subject

That is exactly what you would expect in a sentence like The pomegranate is tasty.

Он refers back to гранат.

Since гранат is a masculine noun, the pronoun is он.

Russian often uses a pronoun in the second clause where English might use it:

• Пусть гранат и вкусный, он быстро пачкает пальцы.

This is natural Russian. You could sometimes omit the pronoun in other contexts, but here using он makes the sentence flow clearly and helps separate the two clauses.

Also remember: Russian nouns have grammatical gender, so it becomes:

• он for masculine nouns
• она for feminine nouns
• оно for neuter nouns

Yes, from the context, it clearly means pomegranate.

That is useful to know because гранат can also mean other things in different contexts, such as:

• garnet
• part of the word for grenade in some forms and compounds

But in a sentence about being tasty and staining your fingers, the meaning is obviously pomegranate.

Пачкать means to dirty, to stain, to make something messy/dirty.

So:

• он быстро пачкает пальцы means
• it quickly stains/makes your fingers messy

Here пачкает is:

• 3rd person singular
• present tense
• imperfective

That fits because this is a general statement about what pomegranates do.

Because пальцы is the direct object of пачкает, and пачкать normally takes the accusative case.

The noun is палец = finger.

Plural forms:

• nominative plural: пальцы
• accusative plural: пальцы

For inanimate masculine nouns in the plural, nominative and accusative are the same, so the form looks unchanged.

So:

• пачкает пальцы = stains fingers

In more natural English, we would usually say stains your fingers, but Russian often leaves out the possessive when it is obvious from context.

Russian often omits possessives like my, your, his, etc. when the meaning is obvious.

So:

• он быстро пачкает пальцы

naturally means something like:

• it quickly stains your fingers
• it quickly stains one’s fingers

Russian does this especially with body parts and things closely connected to the person.

Compare:

• Я мыл руки. = I washed my hands.
• Он закрыл глаза. = He closed his eyes.

So the absence of a possessive is normal.

The word order is natural for a sentence with a concessive first clause:

• Пусть гранат и вкусный, он быстро пачкает пальцы.

The first clause sets up the concession:

• although pomegranate is tasty

Then the main point comes after:

• it quickly stains fingers

Russian word order is flexible, but this order is good for emphasizing contrast: yes, it is tasty, but there is a drawback.

In English, the closest structure would be:

• Although pomegranate is tasty, it quickly stains your fingers.

Yes, Russian often uses a concessive structure together with но in the main clause, though it is not required.

So you could also say:

• Пусть гранат и вкусный, но он быстро пачкает пальцы.

This makes the contrast even more explicit:

• Although pomegranate is tasty, it still quickly stains your fingers.

Without но, the sentence is still perfectly understandable and natural.

It is understandable and correct, but the opening пусть ... и ... sounds a bit more literary, rhetorical, or carefully phrased than the most everyday conversational option.

A more everyday way to say something similar might be:

• Хотя гранат вкусный, он быстро пачкает пальцы.

That said, пусть ... и ... is a real and useful construction, and learners should recognize it.