Learn Hardcore Norwegian: Han glemmer ofte tannlegetimen, men kjøper likevel tanntråd og sjekker om forsikringen dekker tannbehandlingen. - He often forgets the dentist appointment, but still buys dental floss and checks whether the insurance covers the dental treatment.

Norwegian likes to build compound nouns instead of using several separate words.
tannlege = dentist (literally “tooth-doctor”)
time = appointment / lesson / hour
Put together, tannlegetime = dentist appointment.
The -n at the end (tannlegetimen) is the definite article: the dentist appointment.

Why is it Han glemmer ofte tannlegetimen and not Han ofte glemmer tannlegetimen?

In a main clause in Norwegian, the verb must come in second position (the V2 rule).
So the pattern is: Subject – Verb – (Adverb) – Object.
That gives Han glemmer ofte tannlegetimen: He forgets often the dentist appointment.
Han ofte glemmer … sounds wrong because the adverb ofte is incorrectly placed before the verb in a main clause.

What exactly does likevel mean here, and how is it different from allikevel?

likevel means nevertheless / anyway / even so and expresses contrast: he forgets the appointment, but still buys floss and checks the insurance.
allikevel is just a spelling variant of likevel. In modern Norwegian, likevel is more common and recommended, but allikevel is still understood and used, especially informally.
The meaning is the same in this context.

Could I say … men han kjøper likevel tanntråd … instead of … men kjøper likevel tanntråd …?

Yes, both are correct.
Norwegian often drops the repeated subject in the second clause when it is the same as in the first clause: Han glemmer …, men kjøper ….
If you say men han kjøper likevel tanntråd, it is a bit more explicit and slightly heavier in style, but still perfectly natural.

Why is it tanntråd with no article? Could I say en tanntråd?

When you talk about buying or using dental floss as a substance or product in general, you normally treat tanntråd as a mass noun and use it without an article: kjøper tanntråd = buys (some) dental floss.
You could say en tanntråd, but that would sound like one piece/strand of floss, which is unusual in this context.
So kjøper tanntråd is the natural choice here.

Why are tannlegetimen, forsikringen and tannbehandlingen all in the definite form?

The definite forms (-en) signal that these are specific, identifiable things:

tannlegetimen = the dentist appointment (a specific appointment he has)
forsikringen = the insurance (his or a particular insurance policy)
tannbehandlingen = the dental treatment (the particular treatment that might be covered)
Norwegian uses the definite form whenever the speaker and listener both know which concrete thing is being referred to, even if English might leave it generic in some cases.

What does forsikringen come from, and what does it mean?

The base noun is forsikring, which means insurance.
Adding -en gives forsikringen = the insurance.
So om forsikringen dekker … = whether the insurance covers ….

Why is it sjekker om forsikringen dekker … and not sjekker hvis / at forsikringen dekker …?

om here introduces an indirect yes/no question: checks whether the insurance covers….

om = if / whether in indirect questions (Jeg lurer på om …).
hvis = if in conditional sentences (Hvis det regner, … = If it rains, …).
at = that in reported statements (Han sier at … = He says that …).
So om is the correct choice because he is checking whether or not the insurance covers the treatment.

Why is there no preposition after dekker? Why not dekker for tannbehandlingen?

The verb å dekke in the sense to cover (costs) takes a direct object in Norwegian.
So you say forsikringen dekker tannbehandlingen = the insurance covers the dental treatment.
Adding for (dekker for …) is ungrammatical in this meaning.

Why is there a comma before men?

In Norwegian, you normally put a comma between two main clauses that are joined by men (but).
Here we have two main clauses:

Han glemmer ofte tannlegetimen
(men) kjøper likevel tanntråd og sjekker …
Therefore you write: Han glemmer ofte tannlegetimen, men kjøper likevel ….

What is the verb order in om forsikringen dekker tannbehandlingen, and how is it different from main clauses?

om forsikringen dekker tannbehandlingen is a subordinate clause, introduced by om.
In subordinate clauses, Norwegian uses normal Subject–Verb–Object order:
forsikringen dekker tannbehandlingen.
In main clauses, the finite verb must be in second position (V2), but in subordinate clauses you do not apply V2 after the conjunction; you keep S–V–O.

Can I move likevel to other positions, like men likevel kjøper han tanntråd or men han kjøper tanntråd likevel?

Yes, likevel is quite flexible in word order:

… men kjøper likevel tanntråd … (as in the sentence)
… men han kjøper likevel tanntråd …
… men likevel kjøper han tanntråd … (slightly more formal/emphatic)
… men han kjøper tanntråd likevel …
All of these are grammatical; the differences are mostly about rhythm and emphasis, not meaning.

Does ofte apply to all his actions, or only to forgetting the appointment?

In this sentence, ofte clearly belongs only to glemmer: Han glemmer ofte tannlegetimen.
The second part (men kjøper … og sjekker …) describes what he does despite that habit; it is not marked as frequent or infrequent by ofte.
If you wanted ofte to apply to everything, you would have to reformulate the sentence, for example by moving ofte or adding it again in the second clause.