Nasza drużyna ma nowego trenera, a on przychodzi na każdy mecz bardzo wcześnie.

Because nasz changes its form to agree with the noun it describes.

• drużyna is feminine singular
• in the nominative case here
• so nasz becomes nasza

That is the normal Polish agreement pattern:

• nasz trener = our coach
• nasza drużyna = our team
• nasze boisko = our field/pitch

Because nowego trenera is the direct object of ma.

The verb mieć = to have takes the accusative case.
And with masculine animate nouns, the accusative singular looks like the genitive singular.

So:

• nowy trener = nominative, the basic dictionary form
• ma nowego trenera = accusative, after ma

The adjective changes too:

• nowy → nowego
• trener → trenera

This is very common in Polish with masculine animate nouns:

• widzę nowego nauczyciela
• mam dobrego psa
• znam twojego brata

Yes. Nasza drużyna ma nowego trenera literally means Our team has a new coach.

This is a normal way in Polish to say that a team, school, company, etc. now has a certain person in that role.

It does not mean the team physically possesses him. It just means he is their coach.

Here a links two clauses and adds a mild contrast or shift of topic.

So:

• i = and, simple addition
• ale = but, stronger contrast
• a = and/but, often meaning and as for..., while..., or whereas...

In this sentence, a works well because the second clause gives extra information about the new coach:

• Nasza drużyna ma nowego trenera, a on...
• Our team has a new coach, and/as for him, he...

It is not a strong contradiction, so ale would sound too strong.

Yes, Polish often drops subject pronouns, because the verb ending already shows the person.

So you could say:

• Nasza drużyna ma nowego trenera, a przychodzi na każdy mecz bardzo wcześnie.

But on is included here for clarity and emphasis. It helps mark that we are now talking specifically about the coach.

So on is not necessary, but it sounds natural.

Przychodzi is the 3rd person singular present tense of przychodzić.

So:

• ja przychodzę = I come / I am coming
• on przychodzi = he comes / he is coming

In this sentence it describes a habitual action:

• he comes to every match very early

That habitual meaning is especially clear because of na każdy mecz.

Also, przychodzić is imperfective, so it is the normal choice for repeated actions.

Because after a verb of motion like przychodzić, Polish often uses na + accusative to mean to an event.

So:

• na każdy mecz = to every match

Here:

• każdy is accusative singular masculine inanimate
• mecz is masculine inanimate, and its accusative form is the same as the nominative: mecz

Compare the nuance:

• przychodzi na każdy mecz = he comes to every match
• jest na każdym meczu = he is at every match

Both can be correct in different contexts, but this sentence focuses on the act of coming.

Because wcześnie is an adverb, and it modifies the verb przychodzi.

• wcześnie = early
• bardzo wcześnie = very early

You need an adverb here because you are describing how/when he comes, not describing a noun.

Compare:

• wczesny mecz = an early match — adjective
• przychodzi wcześnie = he comes early — adverb

Not completely. Polish word order is fairly flexible, though some versions sound more natural than others.

The original sentence is natural and clear:

• Nasza drużyna ma nowego trenera, a on przychodzi na każdy mecz bardzo wcześnie.

Other possible versions:

• Nasza drużyna ma nowego trenera, a on bardzo wcześnie przychodzi na każdy mecz.
• Nasza drużyna ma nowego trenera, a bardzo wcześnie przychodzi on na każdy mecz.

These alternatives may change the emphasis slightly, but not the basic meaning.

The original word order is a good neutral model for a learner.