Det lykkes begynderen at forstå regelbogen før timeglasset er tomt.

Here det is a formal/dummy subject. Danish often uses this pattern with lykkes:

det lykkes + someone + at + infinitive

So Det lykkes begynderen at forstå ... is literally something like It succeeds for the beginner to understand ..., but natural English would usually make the beginner the subject instead.

Not in the way an English learner might first expect. In this sentence, lykkes is best learned as the normal verb form meaning to succeed / to manage in this construction.

Danish does have -s passives, but some -s forms are just established verbs. So here you should treat at lykkes as its own verb, not as a passive version of some everyday active verb you need to reconstruct.

Not grammatically. The formal subject is det.

But begynderen is the person who experiences the success, so it is the logical or semantic doer in English terms. That is why English would normally say something like the beginner manages to understand..., even though Danish uses the structure with det.

Because begynderen is the definite singular form: the beginner.

• en begynder = a beginner
• begynderen = the beginner

Danish usually adds the definite article to the end of the noun instead of putting a separate word in front.

Because at forstå is an infinitive clause: to understand.

After lykkes in this pattern, Danish uses at + infinitive:

• Det lykkes mig at finde nøglerne.
• Det lykkes begynderen at forstå regelbogen.

So at here is just the infinitive marker, like English to.

Because Danish, like English, makes a lot of compound nouns, and Danish usually writes them as one single word.

So:

• regel = rule
• bog = book
• regelbog = rulebook
• regelbogen = the rulebook

The definite ending goes on the whole compound, not on the individual parts.

Because the nouns have different grammatical genders.

• en bog → bogen
• et timeglas → timeglasset

So:

• -en is the usual definite singular ending for common gender nouns
• -et is the usual definite singular ending for neuter nouns

That is why you get regelbogen but timeglasset.

Because tomt agrees with timeglasset, which is a neuter noun (et timeglas).

Predicate adjectives in Danish often show agreement too:

• en kop er tom = a cup is empty
• et glas er tomt = a glass is empty
• glassene er tomme = the glasses are empty

So timeglasset er tomt is the correct neuter form.

Før means before and introduces a subordinate clause here:

før timeglasset er tomt = before the hourglass is empty

This is a full clause, so Danish uses the finite verb er, not an infinitive. The structure is straightforward here:

• før = conjunction
• timeglasset = subject
• er = verb
• tomt = complement

An English speaker may expect something unusual after before, but this part is actually quite close to English.

Because Danish present tense can describe a current situation, a general statement, or a vivid narrative situation.

So lykkes and er do not have to mean that this is happening right now only. They can also present the situation more generally: the beginner manages to understand the rulebook before the hourglass runs out / is empty.