Learn Hardcore Ukrainian: Мій брат витриваліший за мене.

The form витриваліший is the comparative degree: more enduring / more resilient / having more stamina. So the whole sentence means roughly: My brother has more stamina than I do.

How is the comparative form витриваліший formed from витривалий?

Витривалий → витриваліший follows a common pattern for forming the comparative of many adjectives:

Take the stem: витривал-
Add -іший: витривал
- іший → витриваліший

Other examples with the same pattern:

сміливий (brave) → сміливіший (braver)
гарний (nice) → гарніший (nicer)

So -іший is a regular comparative ending for many two-syllable adjectives ending in -ий.

Why is there no word for is in Мій брат витриваліший за мене?

In Ukrainian, the verb to be (бути) is normally omitted in the present tense in such sentences.

So:

Мій брат витриваліший за мене.
literally: My brother more-resilient than me.

If you include є (the present form of бути), it sounds more formal or emphatic:

Мій брат є витриваліший за мене.

Everyday speech strongly prefers the version without є.

What is the function of за here, and can it always be translated as than?

In this sentence за introduces the second part of a comparison and is best translated as than:

витриваліший за мене → more resilient than me

But за has many other meanings in other contexts (behind, for, in exchange for, etc.), so it is not always translated as than.
It works as than specifically in comparative constructions like:

Він старший за мене. – He is older than me.
Вона розумніша за сестру. – She is smarter than (her) sister.

What case is мене in, and why is it not я?

Мене is in the accusative case (for the pronoun я the accusative and genitive forms are identical: мене).

The preposition за in this comparative use requires the accusative:

за мене – than me (accusative)
за нього – than him (accusative)
за нас – than us (accusative)

So за я is grammatically impossible; the form after за must be мене.

Can I use ніж instead of за here? What is the difference between за and ніж?

Yes, you can say:

Мій брат витриваліший, ніж я.
or
Мій брат витриваліший, ніж я є. (more formal/explicit)

Main points:

ніж behaves like than in English and usually comes after a comma.
After ніж, you normally use the nominative: ніж я, ніж він, etc.
After за, you use the accusative: за мене, за нього.

In everyday speech, both за and ніж are common. ніж can sound a bit more bookish in some contexts, but the difference here is small; both are correct:

Мій брат витриваліший за мене.
Мій брат витриваліший, ніж я.

Does витриваліший change for feminine and plural subjects?

Yes. Comparative adjectives in Ukrainian usually agree with the noun in gender and number. From витривалий:

Masculine singular: витриваліший
- Мій брат витриваліший. – My brother is more resilient.
Feminine singular: витриваліша
- Моя сестра витриваліша. – My sister is more resilient.
Neuter singular: витриваліше
- Моє тіло витриваліше. – My body is more resilient.
Plural (all genders): витриваліші
- Мої друзі витриваліші. – My friends are more resilient.

So витриваліший in the example matches брат (masculine singular).

How would I say My sister is more resilient than me?

You just change брат (brother) to сестра (sister) and match the adjective in gender:

Моя сестра витриваліша за мене.
or
Моя сестра витриваліша, ніж я.

Моя and витриваліша are feminine singular forms, agreeing with сестра.

What is the grammatical role and form of мій in this sentence?

Мій is a possessive pronoun meaning my.

Grammatically, it behaves like an adjective and agrees with брат in:

Gender: masculine
Number: singular
Case: nominative

So we get:

Мій брат – my brother
Feminine: Моя сестра – my sister
Neuter: Моє місто – my city
Plural: Мої друзі – my friends

Would Мій брат витриваліший за я be ever correct?

No, за я is incorrect.

Reasons:

The preposition за here requires the accusative case.
The accusative of я is мене, not я.

So the only correct form with за is:

Мій брат витриваліший за мене.

Can I change the word order to Мій брат за мене витриваліший?

Yes, this word order is possible:

Мій брат за мене витриваліший.

It is still grammatical and understandable. The meaning is the same, but:

Мій брат витриваліший за мене. – more neutral, very natural.
Мій брат за мене витриваліший. – somewhat more expressive; за мене is highlighted in the middle.

In simple statements, learners are safest using the neutral order:

[Subject] [comparative adjective] за [comparison].

How would I say My brother is the most resilient instead of more resilient?

To form the superlative, add the prefix най- to the comparative form:

витриваліший (more resilient) → найвитриваліший (the most resilient)

Examples:

Мій брат найвитриваліший. – My brother is the most resilient.
Мій брат найвитриваліший у родині. – My brother is the most resilient in the family.

Is витриваліший the same as stronger, or is there a nuance difference?

There is a nuance:

витривалий / витриваліший – focuses on endurance, stamina, ability to withstand effort or hardship for a long time.
сильний / сильніший – focuses on physical strength or power (or metaphorical strength).

So:

Мій брат витриваліший за мене. – He can keep going longer; he has better stamina.
Мій брат сильніший за мене. – He is physically stronger than me.

In English you might translate витриваліший as more resilient, has more stamina, or more enduring, depending on context.