Jeg håper at tannbehandlingen ikke blir for dyr.

Yes. at is a subordinating conjunction that works like English “that”:

• Jeg håper at tannbehandlingen ikke blir for dyr.
  = I hope *that the dental treatment doesn’t become / won’t be too expensive.*

Just like in English, “that” is often optional:

• Jeg håper tannbehandlingen ikke blir for dyr.
  (perfectly natural, especially in speech)

So:

• håper at … = hope that …
• at can be dropped in many everyday sentences, but keeping it is never wrong.

Because this is a subordinate clause introduced by at.

Rule of thumb:

• Main clause: verb is in 2nd position, and ikke usually comes after the finite verb.
  
  • Tannbehandlingen blir ikke for dyr.
• Subordinate clause (after at, fordi, hvis, etc.): ikke comes before the finite verb.
  
  • … at tannbehandlingen ikke blir for dyr.

So the pattern in a clause with at is:

at + subject + ikke + finite verb + …
at tannbehandlingen ikke blir for dyr

Norwegian bli means “become / turn out / end up (being)”, and it’s very commonly used to talk about future or eventual outcome:

• Tannbehandlingen blir for dyr.
  = The dental treatment ends up / turns out / will be too expensive.

If you say:

• Tannbehandlingen er for dyr.
  you’re talking more about its current state (as if you already know the price and are judging it now).

With håper, you’re talking about a future or unknown result, so blir is more natural:

• Jeg håper at tannbehandlingen ikke blir for dyr.
  = I hope the treatment won’t be / doesn’t end up being too expensive.

The -en ending makes the noun definite (“the …”), like a built-in article.

• tannbehandling = dental treatment (in general)
• tannbehandlingen = the dental treatment (a specific one we both know about)

In this sentence, the speaker is clearly referring to a specific, upcoming treatment (for example, the visit you’ve booked), so Norwegian prefers the definite form:

• Jeg håper at tannbehandlingen ikke blir for dyr.
  = I hope the (planned) dental treatment won’t be too expensive.

It’s one compound noun with a definite ending:

• tann = tooth
• behandling = treatment
• tannbehandling = dental treatment
• tannbehandlingen = the dental treatment (definite, singular)

In Norwegian, compounds are written as one word, and the definite ending (-en / -et / -a) is added to the last part of the compound:

• barnerom → barnerommet (the children’s room)
• tannbehandling → tannbehandlingen (the dental treatment)

for before an adjective usually means “too (much)” in a negative, “more than desirable” way.

• for dyr = too expensive (more than I can or want to pay)
• veldig dyr / svært dyr = very expensive (just describing high price, not necessarily unacceptable)

So:

• Jeg håper at tannbehandlingen ikke blir for dyr.
  = I hope the treatment doesn’t become *too expensive (i.e., more than I can afford).*

Yes, you can drop at here, and the word order stays the same:

• Jeg håper at tannbehandlingen ikke blir for dyr.
• Jeg håper tannbehandlingen ikke blir for dyr.

In spoken Norwegian, many people leave out at after common verbs like:

• jeg tror (at) … – I think (that) …
• jeg vet (at) … – I know (that) …
• jeg håper (at) … – I hope (that) …

But even when at is omitted, the following clause is still treated like a subordinate clause, so you keep:

subject + ikke + verb
tannbehandlingen ikke blir …

You can say that, but it’s usually not necessary, and often sounds heavier or more formal.

Natural options:

• Jeg håper at tannbehandlingen ikke blir for dyr.
• Jeg håper tannbehandlingen ikke blir for dyr.

Adding vil or skal:

• … ikke vil bli for dyr – can sound more like a prediction or possibility.
• … ikke skal bli for dyr – can sound like a plan or intention.

In everyday speech, blir alone usually covers the idea “will be / ends up being.” So the original sentence is the most typical.

They’re different:

• tannbehandling = dental treatment / dental work (the procedure itself)
• tannlegen = the dentist (the person)
• tannlege = a dentist (profession)

So:

• Jeg skal til tannlegen.
  = I’m going to the dentist.
• Jeg skal få tannbehandling.
  = I’m going to get dental treatment.

In Jeg håper at tannbehandlingen ikke blir for dyr, you are talking about the treatment, not the dentist.

Yes, that’s also grammatically correct, but it changes the emphasis a bit.

• … at tannbehandlingen ikke blir for dyr.
  = neutral, standard word order.
• … at ikke tannbehandlingen blir for dyr.
  = slight emphasis on “not the *treatment (that) becomes too expensive”* — for example, contrasting it with something else that might be expensive.

For a neutral, everyday sentence, tannbehandlingen ikke blir is the usual choice.