Hudba je můj život.

Because můj has to agree with the noun it describes, and that noun is život.

• život is masculine singular
• so the correct form is můj

Compare:

• můj život = my life
• má hudba = my music
• moje auto = my car

It is in the nominative case. After být in a simple identifying sentence, Czech often uses the nominative for the predicate noun, so Hudba je můj život is natural.

You may also see Hudba je mým životem, where mým životem is instrumental. That version is possible too, but here it sounds more formal or stylistically marked.

Czech word order is flexible, but it changes the emphasis.

• Hudba je můj život is the natural, idiomatic way to say that music means everything to you.
• Můj život je hudba is grammatical, but it sounds less natural for that usual meaning and shifts the focus to my life.

So the original sentence is the better choice in most situations.

Because Czech nouns have grammatical gender. Every noun belongs to a gender category, and that affects pronouns, adjectives, and sometimes verb forms.

• hudba is feminine
• život is masculine

You usually need to learn the gender together with the noun.

A good rough guide is mooy.

• ů sounds like the oo in food
• j sounds like English y in yes

So můj is approximately mooy, said as one syllable.

A rough guide is:

HUD-ba ye mooy ZHI-vot

More specifically:

• Hudba = HUD-ba
• je = ye
• můj = mooy
• život = ZHI-vot

Also:

• ž sounds like the s in measure
• Czech stress normally falls on the first syllable of each word

Normally, no. In standard Czech, you usually keep the present-tense form of být. So Hudba můj život would sound incomplete.

You might see the verb omitted in poetry, headlines, or very special stylistic contexts, but not in an ordinary sentence like this.

Sometimes, yes, but it changes the tone.

• hudba is the neutral, standard word for music
• muzika is more colloquial or stylistically colored

So for a simple standard sentence, hudba is the safest and most natural choice.