sensei ha tugi no siken ha kinyou da to iimasita.

Because there are really two layers:

• Main sentence: 先生は … 言いました
• Quoted content: 次の試験は金曜だ

The first は marks 先生 as the topic of the whole sentence.
The second は marks 次の試験 as the topic inside what the teacher said.

So the structure is roughly:

• As for the teacher,
• as for the next exam, it is Friday,
• he/she said.

This can look strange to English speakers, but it is normal in Japanese when one clause contains another.

Using は makes 先生 the topic: we are talking about the teacher and reporting what the teacher said.

If you used が instead:

• 先生が次の試験は金曜だと言いました

it would put more focus on the teacher as the one who said it, possibly contrasting with someone else.

Very roughly:

• 先生は = as for the teacher
• 先生が = the teacher is the one who said it

Both can be correct, but the nuance changes.

次 means next and 試験 means exam/test.

The の connects them, so:

• 次の試験 = the next exam

This is a very common pattern in Japanese:

• 日本の車 = Japanese car / car of Japan
• 明日の予定 = tomorrow’s plans
• 次の試験 = next exam

So here の is acting like a linker between 次 and 試験.

Both mean Friday.

• 金曜 is a shorter form
• 金曜日 is the full form

In everyday Japanese, shortened weekday forms like these are very common:

• 月曜 = Monday
• 火曜 = Tuesday
• 水曜 = Wednesday
• 木曜 = Thursday
• 金曜 = Friday

So 金曜 is not unusual at all. It is just a more compact way to say Friday.

Because 金曜 is the predicate noun here, followed by the copula だ.

So:

• 次の試験は金曜だ = The next exam is Friday

This is different from a sentence like:

• 試験は金曜にあります = The exam is on Friday

In your sentence, 金曜 is not being marked with に because the structure is X は Y だ: X is Y.

Because the quoted sentence before と normally appears in plain form.

Here the quoted content is:

• 次の試験は金曜だ

Since 金曜 is a noun, the plain copula is だ.

Then と marks that whole clause as what was said:

• 次の試験は金曜だ
  • と
    • 言いました

Even though the quoted part is plain, the whole sentence is still polite because the main verb is polite:

• 言いました

This is very common in Japanese: plain form inside the quote, polite form in the main sentence.

と is the quotation particle here.

It marks the content of what someone said. In this sentence, it attaches to:

• 次の試験は金曜だ

So the structure is:

• [quoted content] と 言いました
• [content] said

You can think of it as meaning something like that, although it is not always translated directly.

Without と, the sentence would not sound like standard Japanese.

Because the act of saying happened in the past.

So 言いました means said.

This does not mean that Friday is in the past. It only means that the teacher made that statement earlier.

Japanese often uses past tense this way:

• the speaking happened before now
• the content of the statement may still be true now

So the tense applies to the act of speaking, not necessarily to the exam itself.

The understood subject is 先生.

Japanese often does not mark the subject the same way English does. Here, 先生は sets up the topic, and from context we understand that the teacher is the one who said it.

So even though Japanese is not forcing an explicit English-style subject structure, the meaning is still:

• The teacher said ...

Also, the person the teacher said it to is not mentioned, because Japanese often leaves out information that is not necessary.

Because Japanese usually puts the content before 言う when reporting speech.

So the order is:

• what was said
• と
• said

That is normal Japanese word order.

English often does this too in a way:

• The teacher said that the next exam is Friday

Japanese just keeps the reported content before the verb very consistently, since verbs tend to come later in the sentence.

Yes, it is natural.

A very normal written version would be:

• 先生は、次の試験は金曜だと言いました。

The comma after 先生は helps show the pause, but it is not required.

The double は may look odd at first, but it is fine because:

• one は belongs to the main sentence
• the other は belongs to the quoted statement

So the sentence is not broken or repetitive in a bad way. It is a normal example of nested structure in Japanese.