Learn Hardcore Danish: De træner normalt to gange om ugen, men i denne uge har de sjældent tid. - They normally train twice a week, but this week they rarely have time.

In Danish, most verbs have just one present-tense ending: -r.
The infinitive is at træne; to make the present tense, you normally just add -r: træner.

at træne → de træner (they train)
There is no extra -er ending like in English work → works; Danish doesn’t change for person or number: jeg/du/han/vi/de træner are all the same.

So trænerer is not a valid Danish form.

Can I move normalt to another place, like Normalt træner de to gange om ugen?

Yes. Danish adverbs like normalt are quite flexible in word order.
All of these are grammatically fine, with slightly different emphasis:

De træner normalt to gange om ugen. (neutral; normalt sits in the “middle field”)
Normalt træner de to gange om ugen. (emphasis on normally; contrast with this particular week)
De træner to gange om ugen normalt. (also possible, but sounds a bit less neutral; normalt feels more like an afterthought)

The most typical neutral versions are the first two.

Why is it to gange om ugen and not something like to gange i ugen or to gange per uge?

Om in time expressions often means per or a in English:

en gang om dagen = once a day
to gange om ugen = twice a week

To gange i ugen can be understood, but om ugen is the normal, idiomatic way to say per week.
Per uge exists, but it’s much more formal/technical (e.g. in statistics, contracts, or medicine). In everyday speech, use om ugen.

Why is it i denne uge and not just denne uge?

For time periods like this week / this month / this year, Danish normally uses a preposition:

i denne uge (this week)
i denne måned (this month)
i dette år (this year, more formal; often i år instead)

Saying just denne uge is possible in some structures (e.g. denne uge er travl), but as a time adverbial you would usually include i: i denne uge = during this week.

Could I say den her uge instead of denne uge? Is there a difference?

Yes, you can say:

i denne uge (more neutral/written)
i den her uge (more colloquial/spoken)

Both mean this week.
Den her sounds a bit more informal and is very common in spoken Danish; denne is slightly more formal and common in writing or careful speech.

Why is it har de sjældent tid and not de sjældent har tid?

In a main clause, Danish word order is:

First place: usually a subject or a whole adverbial (here it’s i denne uge that’s been fronted),
Then the finite verb (har),
Then the subject (de),
Then adverbs like sjældent,
Then the rest (tid).

So when i denne uge is first, the correct order is:

i denne uge har de sjældent tid

De sjældent har tid would break normal main-clause word order; it could only appear inside a larger structure (e.g. folk, der sjældent har tid = people who rarely have time).

Why is it sjældent tid and not sjældent har tid after i denne uge?

Here sjældent is an adverb that modifies the verb phrase har tid (have time).
Standard order in a main clause with har as the main verb is:

Subject – har – adverb – object/complement

So:

De har sjældent tid. (They rarely have time.)

When you front i denne uge, that order inside the clause stays the same:

i denne uge har de sjældent tid.

You could say i denne uge har de sjældent haft tid (have rarely had time) if you wanted a past-like meaning, but that changes the tense and nuance.

Is har here an auxiliary (like in English have done) or just have as a normal verb?

Here har is a main verb meaning have / possess time:

De har sjældent tid. = They rarely have time.

It is not forming a perfect tense.
A perfect tense would look like:

De har trænet. (They have trained.)
De har ofte trænet. (They have often trained.)

So in this sentence, har behaves like have in have money, have time, have a car.

Does normalt change form depending on gender or number, like adjectives do?

No. Normalt here is an adverb, not an adjective.
Danish adjectives inflect for gender/number (normal, normalt, normale), but adverbs do not inflect:

Adjective: et normalt problem, en normal uge, normale uger
Adverb: De træner normalt to gange om ugen.

So as an adverb, normalt stays the same; it does not agree with de.

What is the nuance difference between normalt and sjældent in this sentence? They both seem to talk about frequency.

Yes, both are adverbs of frequency, but they describe different situations:

normalt = normally / usually / under normal circumstances
sjældent = rarely / seldom

In the first clause:
De træner normalt to gange om ugen → Under normal circumstances, their routine is twice a week.

In the second clause:
i denne uge har de sjældent tid → In this particular week, time is rare; they don’t often have free time.

So normalt sets the general rule; sjældent describes the exception this week.

Is the comma before men required in Danish: ..., men i denne uge ...?

Yes, in standard Danish punctuation you normally put a comma before men when it connects two main clauses:

De træner normalt to gange om ugen, men i denne uge har de sjældent tid.

You would also put a comma before men between full clauses in other contexts.
In casual writing some people drop commas more often, but the comma here is considered correct and standard.