Learn Hardcore Norwegian: Tannlegen sier at hun bør bruke tanntråd hver kveld. - The dentist says that she should use dental floss every evening.

So you do not say den tannlegen in a neutral sentence; that would sound wrong here. You only use den tannlegen in special cases for extra emphasis or contrast, roughly like saying that dentist in English.

What does sier at mean, and what is the function of at here?

sier means says.
at is a conjunction that introduces a that-clause, just like English that:

Tannlegen sier at hun bør bruke tanntråd hver kveld.
= The dentist says that she should use dental floss every evening.

So the pattern is:

[main clause]
- at
  - [subordinate clause]

In normal written Norwegian you keep at here; in very informal spoken Norwegian, people sometimes drop it (especially in some dialects), but learners should keep at.

What exactly does bør mean? Is it the same as must or should?

bør is a modal verb that normally corresponds to English should or ought to. It expresses advice or a recommendation, not a strict obligation.

hun bør bruke tanntråd = she should use dental floss

Compare with other modal verbs:

må = must / have to (strong necessity)
- Hun må bruke tanntråd. = She must use dental floss.
skal = shall / is supposed to / is going to (plan or stronger expectation)
- Hun skal bruke tanntråd. = She is supposed to use dental floss.

So:

bør = good idea, recommendation
må = need, requirement
skal = plan, arrangement, or strong expectation

Why is it bør bruke and not bør bruker?

In Norwegian, after a modal verb like bør, må, skal, kan, vil, ønsker å, the next verb is in the infinitive form:

bør bruke (should use)
må bruke (must use)
kan bruke (can use)

So:

bør = conjugated modal verb
bruke = infinitive (dictionary form)

Using bruker after bør (bør bruker) would be incorrect.
No å is used between bør and bruke; modals take a bare infinitive:

Jeg bør bruke tanntråd.
Jeg vil bruke tanntråd.
Jeg kan bruke tanntråd.

Why is the pronoun hun used here, and not henne?

hun is the subject form (like English she).
henne is the object form (like English her as an object).

In the sentence:

hun is the subject of the verb bør bruke inside the at-clause:
- … at hun bør bruke tanntråd … = … that she should use dental floss …

You use henne when it is the object:

Jeg ser henne. = I see her.
Tannlegen sier til henne at … = The dentist says to her that …

So here, hun is correct because she is doing the action (using dental floss).

How is tanntråd formed, and does it always mean dental floss?

tanntråd is a compound word:

tann = tooth
tråd = thread

Literally: tooth-thread, which is exactly what dental floss is.

Usage:

tanntråd (no article) is often used like a mass noun, just like dental floss in English:
- Hun bør bruke tanntråd. = She should use dental floss.
If you need to, you can also treat it as a count noun:
- en tanntråd = a (piece / package of) dental floss
  But often in practice people still just say tanntråd.

Why is the time expression hver kveld placed at the end? Could it go somewhere else?

hver kveld means every evening / every night (in the evening).

In Norwegian, time expressions are often placed:

At the end of the clause:
- Hun bør bruke tanntråd hver kveld.
Or at the beginning for emphasis or topic:
- Hver kveld bør hun bruke tanntråd.

Both are correct. The version in your sentence:

… hun bør bruke tanntråd hver kveld.

is the most neutral, everyday word order:
subject – verb – object – time.

Can I move hver kveld inside the clause, like … at hun hver kveld bør bruke tanntråd?

Yes, that is grammatically possible:

Tannlegen sier at hun hver kveld bør bruke tanntråd.

However, this word order sounds a bit more formal or slightly marked in style.
The most natural everyday options are:

Tannlegen sier at hun bør bruke tanntråd hver kveld. (neutral)
Tannlegen sier at hun hver kveld bør bruke tanntråd. (slightly more formal/emphatic)
Hver kveld sier tannlegen at hun bør bruke tanntråd. (emphasis on every evening the dentist says this)

For learners, it is safest to keep hver kveld either at the end of the clause, or at the very beginning of the whole sentence.

Why is it sier (present tense says) and not a past tense like sa (said)? Could Norwegian sier refer to something she said in the past?

sier is the present tense and normally refers to something that is currently true / generally true / happening now:

Tannlegen sier at … = The dentist says (is saying / says in general) that …

If you want a clear past tense, you use sa:

Tannlegen sa at hun bør bruke tanntråd hver kveld.
= The dentist said that she should use dental floss every evening.

Unlike English, Norwegian does not normally use the present tense (sier) for a completed, specific past event; for that, you switch to sa. So:

sier = says / is saying / says (regularly, in general)
sa = said (one or more times in the past)

How do you pronounce difficult parts like tannlegen, sier, hver, and tanntråd?

Approximate pronunciation (standard Eastern Norwegian):

tannlegen: TANN-leh-gen
- Stress on the first syllable tann
- g in -legen is a soft g, not like English go
sier: roughly SEE-er (often [sɪːr], so the second vowel is very weak)
hver: roughly vair
- The h is usually silent; hv is pronounced like v
- æ is like the vowel in English air
tanntråd: TANN-traw(d)
- Stress on tann
- å is similar to the vowel in English bought or law
- Final d can be weak or almost silent depending on dialect

These are only approximations, but they should get you close to how the sentence is normally pronounced.