Installeer de lamp en schroef hem vast.

Installeer is an imperative: a command form.

In Dutch, commands usually use the verb stem without a subject:

• Installeer de lamp.
• Pak het gereedschap.
• Wacht even.

So Dutch often leaves out je in direct instructions, just like English says Install the lamp rather than You install the lamp.

Because lamp is a de-word in Dutch.

Dutch nouns take either de or het, and this has to be learned with the noun:

• de lamp
• de tafel
• het huis
• het raam

A useful extra detail: the diminutive would be het lampje, because all diminutives take het.

Because lamp is a singular de-word, and Dutch often uses hij/hem to refer back to singular de-words that are things.

So:

• de lamp → hem
• het huis → het

That said, this area is a bit messy in real Dutch. Some nouns are traditionally feminine, and you may also hear ze/haar for some things, especially depending on region or style. But hem for an inanimate singular de-word is very common, especially in standard Dutch from the Netherlands.

This is a normal Dutch spelling pattern.

The verb is schroeven. Its stem is pronounced with a final f sound, and Dutch spelling reflects that at the end of a syllable:

• schroeven → schroef
• leven → leef
• durven → durf

So in the imperative you get schroef.

It is based on the separable verb vastschroeven.

So the dictionary form is:

• vastschroeven = to screw fast / to screw into place

But in a main clause or an imperative, Dutch splits it:

• Schroef hem vast.

This is very common in Dutch:

• opbellen → Bel me op.
• aandoen → Doe het licht aan.
• vastmaken → Maak het vast.

Vast adds the idea of firmly, securely, or in place.

So schroeven by itself is just to screw, while vastschroeven means something like:

• screw it tight
• screw it into place
• fasten it with screws

Dutch often uses little particles like vast, aan, af, op, and uit to create more specific meanings.

Because vastschroeven is separable.

In a main clause or command, the finite verb comes earlier, and the separated particle goes to the end:

• Schroef hem vast.
• Doe de deur dicht.
• Zet het licht aan.

But in the infinitive, it stays together:

• Je moet hem vastschroeven.

So both forms are connected:

• vastschroeven
• schroef ... vast

Because the object normally comes before the separated particle in this kind of sentence.

So Dutch prefers:

• Schroef hem vast. not
• Schroef vast hem.

You can think of the separable particle as something that tends to stay near the end of the clause, while the object sits before it.

Yes, absolutely.

Both are correct:

• Installeer de lamp en schroef de lamp vast.
• Installeer de lamp en schroef hem vast.

Using hem is just more natural because it avoids repeating de lamp. Once the noun has already been mentioned, Dutch usually switches to a pronoun if the meaning is still clear.