Na het flauwvallen bleef hij een uur uitrusten.

Both are possible, but they work a little differently.

• na het flauwvallen = after fainting / after the fainting
• nadat hij flauwviel = after he fainted

In na het flauwvallen, the action is turned into a noun-like expression. Dutch does this very often with het + infinitive after prepositions such as na, voor, bij, and tijdens.

So:

• Na het eten = after eating
• Na het flauwvallen = after fainting

This version is compact and very natural.

Here, flauwvallen is being used as a noun-like infinitive, and in Dutch that often takes het.

So:

• het eten = eating / the eating
• het zwemmen = swimming
• het flauwvallen = fainting

This is similar to English the eating or eating, although English usually does not need an article in the same way.

In the infinitive, it is written as one word: flauwvallen.

But it is a separable verb, so in some forms it splits:

• infinitive: flauwvallen
• present: hij valt flauw
• past: hij viel flauw
• past participle: flauwgevallen

So in het flauwvallen, it stays together because it is an infinitive.

blijven + infinitive often means to keep doing something, to remain doing something, or to stay doing something.

So:

• Hij rustte een uur uit = He rested for an hour.
• Hij bleef een uur uitrusten = He stayed/rested on for an hour; the idea is that the resting continued.

The version with bleef ... uitrusten gives a stronger sense of duration or continuation after the fainting.

Because Dutch main clauses normally place the finite verb in second position, and other verbs go later, often at the end.

Here the finite verb is bleef:

• Na het flauwvallen | bleef | hij | een uur | uitrusten

So:

• bleef is in second position
• uitrusten stays at the end

This is very normal Dutch word order.

Yes. uitrusten is separable.

Examples:

• infinitive: uitrusten
• present: hij rust uit
• past: hij rustte uit
• with another verb: hij bleef uitrusten

So just like flauwvallen, it is together in the infinitive, but it can split when conjugated.

In Dutch, lengths of time are often used directly without a preposition.

So:

• een uur = for an hour
• twee dagen = for two days
• de hele middag = all afternoon

Dutch does not need a separate word for for here.

Yes, absolutely. That is a very natural alternative.

The difference is mostly in emphasis:

• Na het flauwvallen rustte hij een uur uit
  = After fainting, he rested for an hour.
• Na het flauwvallen bleef hij een uur uitrusten
  = After fainting, he continued/stayed resting for an hour.

The second version highlights the ongoing nature of the resting a bit more.

Because hij is the subject of the verb bleef.

In this sentence, he is the person who did the resting, so Dutch uses the subject form:

• hij = he
• hem = him

Compare:

• Hij bleef uitrusten. = He kept resting.
• Ik zag hem uitrusten. = I saw him resting.

Yes. Dutch allows that too.

For example:

• Hij bleef na het flauwvallen een uur uitrusten.

That is also grammatical. Putting Na het flauwvallen at the beginning gives it a little more emphasis as the starting context of the sentence.

Usually it corresponds most directly to to faint, though in many contexts to pass out can also fit.

A useful point for learners is that Dutch often uses flauwvallen in ordinary, neutral speech. It is not especially formal.

Related forms:

• hij valt flauw = he faints / passes out
• hij is flauwgevallen = he fainted / has fainted / passed out

Yes, here uitrusten means to rest, to recover by resting, or to take it easy for a while.

It often suggests resting in order to regain strength or energy.

So in this sentence, it fits very naturally after fainting: he spent an hour recovering/resting.

Not always. Grammatically it behaves like the fainting, but in natural English you will often translate it more smoothly as just fainting.

So:

• Na het flauwvallen
  natural English: After fainting less natural but closer structurally: After the fainting

This is a good example of where understanding the Dutch grammar matters more than translating word for word.