Behozhatom a táskámat a vizsgára?

It breaks down like this:

• be- = in, into
• hoz = bring
• -hat/-het = can / may
• -om = I with a definite object

So behozhatom literally means something like I can bring it in or may I bring it in.

In this sentence, the it is spelled out as a táskámat = my bag.

The prefix be- adds the idea of going in / bringing in / taking inside.

So:

• hozni = to bring
• behozni = to bring in

Without be-, the sentence would be more general: Can I bring my bag?
With be-, it clearly means Can I bring my bag in?

Because Hungarian verbs often have two kinds of endings:

• indefinite conjugation
• definite conjugation

Here the object is definite: a táskámat = my bag, a specific bag.
That is why the verb uses the definite ending -om.

Compare:

• Behozhatok egy táskát? = Can I bring in a bag?
  (indefinite: a bag)
• Behozhatom a táskámat? = Can I bring in my bag?
  (definite: my bag)

Because táskám means my bag, but here it is the direct object of the verb, so it needs the accusative.

• táska = bag
• táskám = my bag
• táskámat = my bag as the thing being brought

So the -t marks the object, but because of Hungarian sound patterns, the full form becomes táskámat, not just táskám-t.

That is normal in Hungarian. A possessed noun can still take the definite article.

So:

• a táskám = my bag
• a táskámat = my bag (object form)

English usually does not say the my bag, but Hungarian often does use the article with possessed nouns.

It is:

• vizsga = exam
• -ra/-re = a case ending meaning roughly onto / to / for

So a vizsgára means something like:

• to the exam
• for the exam
• in this context, effectively into the exam setting / exam room

This is one of those places where Hungarian uses a case ending where English uses a preposition.

Because a points to a specific exam.

So a vizsgára is like to the exam or for the exam in a specific situation.
In a real context, this usually means the exam I’m about to take.

Hungarian often uses the definite article in places where English may leave it out.

It can mean either, depending on context.

• ability: can
• permission: may

In this sentence, because it is about exam rules, it most naturally means permission:

May I bring my bag into the exam?

But grammatically the same form is used for both ideas.

No, Hungarian word order is quite flexible. It often changes to show focus or emphasis.

The sentence:

Behozhatom a táskámat a vizsgára?

is a natural neutral way to say it.

But other orders are possible, with different emphasis, for example:

• A táskámat behozhatom a vizsgára?
  Emphasis on my bag
• A vizsgára behozhatom a táskámat?
  Emphasis on the exam

So the order is not random, but it is not as fixed as in English.

Yes, very possibly.

Both behozni and bevinni can mean to bring/take in, and in many situations they are close in meaning.

Very roughly:

• hozni = bring (often toward the speaker or reference point)
• vinni = take / carry

So:

• Behozhatom... ?
• Bevihetem... ?

can both sound natural, depending on the situation. In everyday speech, either may be used for asking whether something is allowed inside.

Because Hungarian usually includes the subject inside the verb ending.

In behozhatom, the ending already tells you the subject is I.

So Hungarian often does not need én.

You could say:

• Én behozhatom a táskámat a vizsgára?

but that would add emphasis, something like:

• Can I bring my bag in?
• Am I allowed to?

The plain sentence without én is the normal neutral form.