Мій двоюрідний брат молодший за мене, але дорослішає дуже швидко, як і мій племінник.

Because брат is a masculine singular noun, so the possessive adjective and the descriptive adjective both have to agree with it:

• мій = my for a masculine singular noun
• двоюрідний = masculine singular form to match брат

If the noun were feminine, the forms would change. For example:

• моя двоюрідна сестра = my female cousin

It means male cousin.

Ukrainian often expresses cousin as:

• двоюрідний брат = male cousin
• двоюрідна сестра = female cousin

Literally, двоюрідний is something like second-kin / second-degree related, but in normal usage the whole phrase simply means cousin.

Because молодший is the comparative form: younger.

• молодий = young
• молодший = younger

So:

• Мій брат молодий. = My brother is young.
• Мій брат молодший за мене. = My brother is younger than me.

This is like English young versus younger.

After a comparative adjective like молодший (younger), Ukrainian commonly uses за + accusative to mean than.

So:

• молодший за мене = younger than me

Here мене is the accusative form of я.

This is a very common pattern:

• старший за мене = older than me
• вищий за мене = taller than me
• кращий за мене = better than me

In this sentence, yes, functionally it does.

The preposition за has several meanings in Ukrainian depending on context, but after a comparative it often corresponds to English than:

• молодший за мене = younger than me

So even if you know за from other meanings, here you should understand it as part of the comparison pattern.

Yes. Both are possible.

Two common ways to make this comparison are:

• молодший за мене
• молодший, ніж я

Both mean younger than me.
The version with за + accusative is very common and natural.

дорослішає is the 3rd person singular present tense form of дорослішати.

So it means:

• he/she is growing up
• he/she is maturing
• he/she is becoming more adult

In this sentence, the subject is мій двоюрідний брат, so дорослішає means he is growing up / maturing.

Because дорослішати is specifically about growing up or becoming more mature, not just the passage of time.

So:

• старіти = to grow old / age
• дорослішати = to grow up / become more mature

In this sentence, the idea is not simply that the cousin is aging, but that he is developing and maturing quickly.

Here як і means something like just like or as does.

So:

• ..., але дорослішає дуже швидко, як і мій племінник.

means:

• ..., but he is growing up very quickly, as is my nephew.

Important: this phrase refers to the idea of growing up very quickly, not necessarily to the entire first clause. In other words, it means the nephew also matures quickly.

Because Ukrainian often leaves out words that are already understood from the context.

The full idea is:

• ..., але дорослішає дуже швидко, як і мій племінник дорослішає дуже швидко.

But repeating the whole verb phrase would sound heavy and unnecessary, so Ukrainian omits it.

English does something similar:

• He is growing up quickly, and so is my nephew.

племінник is a masculine singular noun meaning nephew.

That is why it also uses мій:

• мій племінник = my nephew

Again, the possessive adjective agrees with the masculine noun.

The comma before але is standard because але means but and joins two clauses.

• ..., брат молодший за мене, але дорослішає дуже швидко ...

The comma before як і мій племінник separates the added comparison-like phrase:

• ..., дорослішає дуже швидко, як і мій племінник.

So the punctuation helps show the sentence structure clearly.

Not completely. Ukrainian word order is fairly flexible, but this version is very natural.

• дорослішає дуже швидко = is growing up very quickly

You might also see different word orders for emphasis, but the given order is neutral and common. The adverb phrase дуже швидко simply modifies the verb дорослішає.