Teklifte kayıt ücreti ve geçerlilik süresi yazıyor.

Teklifte means in the offer / in the quote / on the proposal.

It is made of:

• teklif = offer, proposal, quotation
• -te = the locative suffix, meaning in / on / at

In business Turkish, teklif very often means a price quote or offer document, not just an abstract proposal.

Because of consonant assimilation.

The locative suffix has four main forms:

• -da
• -de
• -ta
• -te

After a voiceless consonant like f, s, t, k, ç, ş, h, p, Turkish uses t instead of d.

Since teklif ends in f, the suffix becomes -te:

• teklif + de → teklifte

The e part comes from vowel harmony.

It is an indefinite noun compound.

• kayıt = registration
• ücret = fee
• ücreti = fee + compound ending

So kayıt ücreti means registration fee.

This is a very common Turkish pattern:

• first noun stays plain
• second noun takes a suffix such as -ı / -i / -u / -ü

Examples:

• öğrenci kartı = student card
• şehir merkezi = city center
• kayıt ücreti = registration fee

Even though that ending can look like a possessive ending, in this structure it is just part of the normal noun-compound pattern.

Because kayıt ücreti is the normal way to say registration fee as a general label.

Compare:

• kayıt ücreti = registration fee
• kaydın ücreti = the fee of the registration / the fee for your specific registration

So:

• kayıt ücreti is a general category name
• kaydın ücreti refers to a specific registration

English often uses noun + noun, and Turkish often uses this compound structure instead.

It is another noun compound:

• geçerli = valid
• geçerlilik = validity
• süre = duration, period
• süresi = period + compound ending

So geçerlilik süresi means validity period or period of validity.

Again, this is the same compound pattern as kayıt ücreti:

• first noun: geçerlilik
• second noun: süresi

Because Turkish often uses third person singular with inanimate subjects, even when more than one thing is listed.

So this is natural:

• Kayıt ücreti ve geçerlilik süresi yazıyor.

Using yazıyorlar here would sound unnatural in most contexts.

This is especially common when the meaning is something like:

• it says
• it is written
• it appears there

Not in the literal sense.

In this sentence, yazıyor means something like:

• is written
• appears
• it says

Turkish often uses yazıyor this way when talking about information shown in a text, ad, form, website, or document.

For example:

• Kapıda açık yazıyor. = It says open on the door.
• Listede adı yazıyor. = Their name is written on the list.

So here it means that the offer document contains that information.

Both are possible, but they feel slightly different.

• yazıyor = a very natural everyday way to say it says / it is written
• yazılı = more static or formal, like written / in written form

So both of these work:

• Teklifte kayıt ücreti ve geçerlilik süresi yazıyor.
• Teklifte kayıt ücreti ve geçerlilik süresi yazılı.

The version with yazıyor is very common in spoken and practical written Turkish.

Yes. Turkish word order is flexible.

This sentence starts with teklifte to set the scene first:

• In the offer, ...

But you could also say:

• Kayıt ücreti ve geçerlilik süresi teklifte yazıyor.

That still works, but the original version sounds very natural because Turkish often puts the location or context early in the sentence.

It is natural and normal.

Depending on style, you might also see:

• Teklifte kayıt ücreti ve geçerlilik süresi yer alıyor.
  = The offer includes the registration fee and validity period.

• Teklifte kayıt ücreti ve geçerlilik süresi belirtiliyor.
  = The registration fee and validity period are stated in the offer.

• Teklifte kayıt ücreti ile geçerlilik süresi belirtilmiştir.
  = A more formal written style.

But the original sentence is completely natural and easy to understand.