Leg de handdoek op het bed.

Leg is the imperative singular of leggen.

• leggen = to lay / to put something down
• ik leg = I lay / I put
• jij legt = you lay / you put
• leg! = lay! / put!

So in this sentence, leg is the command form used when telling one person what to do.

Because this is a command. In Dutch, commands usually start with the verb.

• Leg de handdoek op het bed. = command
• Je legt de handdoek op het bed. = statement

So the verb-first order is normal for the imperative.

In Dutch imperatives, the subject you is usually left out, just like in English:

• Put the towel on the bed.
• Leg de handdoek op het bed.

Dutch does have ways to make the subject explicit, but in a normal command it is usually omitted.

If you want to be more formal, Dutch often uses:

• Legt u de handdoek op het bed.
  or more naturally:
• Wilt u de handdoek op het bed leggen?

Dutch nouns take either de or het as their definite article.

• de handdoek = the towel
• het bed = the bed

This is something you usually have to learn with each noun.

A useful pattern:

• de is used for most nouns
• het is used for neuter nouns

So handdoek is a de-word, and bed is a het-word.

Because bed is a het-word, so its definite article is het.

That stays true after a preposition:

• het bed = the bed
• op het bed = on the bed

The preposition op does not change the article. It just comes before the noun phrase.

Dutch often chooses different put/place verbs depending on the position of the object.

• leggen is used when something is placed in a lying / flat / horizontal position
• zetten is used when something is placed standing upright
• doen can sometimes mean put, but it is more general and often less precise

A towel is normally thought of as something you lay down, so leggen is the natural choice.

Examples:

• Leg het boek op tafel. = Put the book on the table.
• Zet het glas op tafel. = Put the glass on the table.

Here op means on.

In sentences like this, Dutch uses op where English often says on or onto:

• Leg de handdoek op het bed.

The movement idea is mainly shown by the verb leggen. Dutch does not need a separate word like onto here.

So:

• op = on
• with leggen, the full idea becomes put/lay ... on

Yes. Neer can be added to make the idea of putting something down more explicit.

• Leg de handdoek op het bed.
• Leg de handdoek op het bed neer.

Both are natural. The version without neer is already complete and very common.
Adding neer can sound a bit more explicit or slightly more conversational.

Usually, no—not in this sentence.

Op bed is an idiomatic expression that usually means in bed / to bed, especially with people:

• Ik lig op bed. = I’m in bed.
• Hij gaat op bed. = He goes to bed.

But with a towel, if you mean the physical bed as an object, you normally say:

• op het bed

So Leg de handdoek op het bed is the correct version here.

A normal statement would usually be:

• Je legt de handdoek op het bed.

That means you put the towel on the bed or you are putting the towel on the bed, depending on context.

So the difference is mainly:

• Leg ... = command
• Je legt ... = statement

Leg is pronounced roughly like English lekh, but the final sound is not a normal English g.

A few points:

• e in leg is a short vowel, like the e in bed
• the final g in standard Dutch is a throaty sound, like the ch in Scottish loch or German Bach

So it is approximately:

• leg → lekh

The exact sound depends on region:

• In the north of the Netherlands, it is often a stronger throaty sound
• In the south and in Flanders, it is often softer

Yes.

In Leg de handdoek op het bed:

• Leg = verb
• de handdoek = direct object
• op het bed = prepositional phrase showing location/destination

So the thing being moved is de handdoek, and the place where it should go is op het bed.