De klimmer draagt een klimgordel en een klimtouw.

Klimmer is a noun for a person, and nouns referring to people are usually de-words in Dutch. Also, many nouns ending in -er are de-words.

So:

• de klimmer = correct
• het klimmer = incorrect

Draagt is the present tense form of dragen.

Dragen can mean:

• to wear
• to carry

In this sentence, English might translate it as wears for the harness, but Dutch uses dragen naturally for both the harness and the rope in one sentence.

Because de klimmer is singular, the verb must also be singular.

The verb is dragen = to wear / to carry.
Its present tense forms are:

• ik draag
• jij/u draagt
• hij/zij/het draagt
• wij/jullie/zij dragen

Since de klimmer is like he/she, you use draagt.

Because Dutch normally uses the indefinite article with each separate singular noun in a list like this.

So:

• een klimgordel
• een klimtouw

That is completely natural Dutch. It works the same way as English a climbing harness and a climbing rope.

Because Dutch normally writes compound nouns as a single word.

So:

• klim
  • gordel → klimgordel
• klim
  • touw → klimtouw

This is very common in Dutch. English often writes similar ideas as separate words, but Dutch usually combines them into one noun.

In Dutch compounds, the first part is often a stem or basic form, not the full infinitive.

So:

• verb: klimmen = to climb
• compound element: klim-

That is why you get:

• klimgordel
• klimtouw

not:

• klimmengordel
• klimmemtouw

This is a normal main clause, so the finite verb comes in the second position.

The structure is:

• De klimmer = subject
• draagt = finite verb
• een klimgordel en een klimtouw = object(s)

So it follows the usual Dutch main-clause pattern: subject + finite verb + rest

No. Even though both use een here, they do not have the same definite article.

They are:

• de klimgordel
• het klimtouw

This is because in Dutch compounds, the gender usually comes from the final part of the word:

• gordel is a de-word → de klimgordel
• touw is a het-word → het klimtouw

You cannot see that difference in the sentence because een is used for both de-words and het-words.

Grammatically, yes, but the meaning becomes less specific.

• gordel can mean belt
• touw just means rope

By using:

• klimgordel
• klimtouw

the sentence clearly refers to climbing equipment.

Draagt is natural and correct. It is a good general verb here.

Depending on context, Dutch could also use other phrasing, for example:

• De klimmer heeft een klimgordel aan. = The climber is wearing a climbing harness.
• De klimmer heeft een klimtouw bij zich. = The climber has a climbing rope with him/her.

But your original sentence is perfectly normal, especially as a simple descriptive sentence.

In many varieties of Dutch from the Netherlands, g is pronounced as a throaty, guttural sound, unlike the English g in go.

So in gordel, the g is usually not an English-style hard g.
In Belgian Dutch and in some southern accents, this sound is often softer.

For a learner, the important point is:

• do not pronounce it like English g in go
• it is usually a more friction-like sound made in the back of the mouth

The plural forms are:

• de klimmer → de klimmers
• de klimgordel → de klimgordels
• het klimtouw → de klimtouwen

Notice that:

• plural nouns take de
• touw becomes touwen in the plural, so klimtouw becomes klimtouwen