Öğretmen tahtaya tebeşirle yazdı, ben de not kâğıdına aynısını geçirdim.

Because with the verb yazmak in this meaning, Turkish usually uses the dative for the surface you write onto: tahtaya yazmak = to write on/onto the board.

• tahta = board
• tahta-ya = to/onto the board

Even though English says on the board, Turkish treats it more like a destination surface.

Compare:

• Tahtaya yazdı. = He/she wrote on the board.
• Tahtada yazı var. = There is writing on the board.

So tahtaya is the natural choice here.

It is the dative case ending, which often means to, toward, or onto.

Since tahta ends in a vowel, Turkish inserts a buffer consonant y before the suffix:

• tahta + a → tahtaya

This buffer y is very common in Turkish when a suffix beginning with a vowel is added to a word that already ends in a vowel.

Yes. Tebeşirle is the shortened form of tebeşir ile.

Both mean with chalk.

• ile = with/by means of
• tebeşir ile → tebeşirle

This suffix-like form -le/-la is extremely common in everyday Turkish:

• kalemle = with a pen
• arabayla = by car
• tebeşirle = with chalk

So tebeşirle yazdı means wrote with chalk.

Here de means also / too.

So ben de means I too or and I also.

In this sentence, it connects the second action to the first one:

• The teacher wrote it,
• and I also copied it down.

Important: this de is written separately because it is the particle meaning also/too.

Compare:

• Ben de geldim. = I came too.
• Evdeyim. = I am at home.
  Here -de is a case ending, not the separate word de.

Because this particle follows vowel harmony.

After a word whose last vowel is front, you usually get de:

• ben de
• sen de

After a word whose last vowel is back, you usually get da:

• o da
• onlar da

So ben de is the correct form.

Because not kâğıdı is a noun compound in Turkish.

• not = note
• kâğıt = paper
• not kâğıdı = note paper / note sheet

In this kind of compound, the second noun usually takes a possessive-looking ending:

• kâğıt → kâğıdı

Then, when you add the dative case, it goes onto the whole compound:

• not kâğıdı + na → not kâğıdına

That n is a buffer consonant used after this type of ending.

So:

• not kâğıdına geçirmek = to copy/write something onto one’s note paper

For the same reason as tahtaya: it marks the place or surface something is transferred onto.

Here the idea is:

• the teacher wrote something onto the board
• I copied the same thing onto my note paper

So Turkish uses the dative again:

• tahtaya
• not kâğıdına

Aynısını means the same one / the same thing / exactly the same in this sentence.

It comes from:

• aynı = same
• aynısı = the same one / the same thing
• aynısını = the same one/thing (as a definite direct object)

The -ı / -i / -u / -ü ending here is the accusative case, used because this is a specific thing being copied.

So:

• aynısını geçirdim = I copied the same thing down

Because aynı by itself is often an adjective:

• aynı kitap = the same book
• aynı cümle = the same sentence

But here it stands alone as a pronoun-like word meaning the same thing. Turkish usually uses aynısı for that:

• Aynısı güzel değil. = The same one is not nice.
• Aynısını istiyorum. = I want the same one.

So in your sentence, aynısını is the natural form because it means that same thing as a specific object.

That is a very common question, because geçirmek has several meanings.

Here, bir şeyi bir yere geçirmek means to copy something onto something else, to write it down, or to transfer it in writing.

So in this sentence:

• aynısını geçirdim = I copied the same thing down

This is similar to expressions like:

• deftere geçirmek = to copy into a notebook
• temize geçirmek = to write out neatly/fair-copy

So this is not the pass through meaning. It is the transfer in writing meaning.

They are both in the simple past tense.

• yaz-dı
  yaz- = write
  -dı = past
  → he/she wrote

• geçir-di-m
  geçir- = transfer/copy/write down
  -di = past
  -m = I
  → I copied / I transferred

So the difference is mainly the subject:

• yazdı = he/she wrote
• geçirdim = I copied

Yes, Turkish could leave it out, because the verb geçirdim already shows I.

So this would also be possible:

• Öğretmen tahtaya tebeşirle yazdı, not kâğıdına aynısını geçirdim.

But ben de adds a bit of contrast or emphasis:

• the teacher did that, and I did my part too

So including ben is natural here because it helps set up the contrast between öğretmen and ben.

Not completely, but the given order is very natural.

Turkish often puts the verb at the end, and other elements before it:

• Öğretmen tahtaya tebeşirle yazdı
• ben de not kâğıdına aynısını geçirdim

You can move parts around for emphasis, but the current version is clear and standard.

For example, Turkish could also say:

• Öğretmen tebeşirle tahtaya yazdı
• Ben de aynısını not kâğıdına geçirdim

Those are still grammatical, but the emphasis changes slightly. The original sentence sounds smooth and normal.