der Haken
hook
A curved object made of metal or plastic that is used to hang up or fasten something.
Häng deine Jacke bitte an den Haken an der Wand.
Please hang your jacket on the hook on the wall.

hook / catch
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hook
A curved object made of metal or plastic that is used to hang up or fasten something.
Häng deine Jacke bitte an den Haken an der Wand.
Please hang your jacket on the hook on the wall.

The figurative meaning is very often used in the question 'Wo ist der Haken?' (What's the catch?). You use it to ask about the hidden disadvantage in an offer that sounds too good to be true.
Ein kostenloses Smartphone? Da muss es doch einen Haken geben.
A free smartphone? There must be a catch.
When referring to the physical object, it is almost always combined with the verb 'hängen' (to hang) and the preposition 'an' (on/at).
Kannst du meinen Mantel bitte an den Haken hängen?
Can you please hang my coat on the hook?
In the meaning of 'Häkchen' (check mark), the verb 'machen' (to make) or 'setzen' (to place/put) is often used. You make or put a check mark after a task on a list.
Ich habe alle Punkte erledigt und überall einen Haken gemacht.
I completed all the items and put a check mark everywhere.
The meaning 'Boxschlag' (boxing punch) is very specific. It is used exclusively in the context of sports, especially boxing or martial arts.
Er beendete den Kampf mit einem gezielten Haken.
He ended the fight with a well-aimed hook.
problem
'das Problem' and 'der Haken' can both refer to a difficulty. 'Problem' is general and can be big or small. 'Der Haken' usually means a disadvantage or a hidden difficulty in something that looks good at first. The phrase 'Der Haken ist ...' (the catch is ...) is especially typical. If you are simply talking about a difficulty, 'Problem' fits. If you want to say that there is a downside to a good offer or plan, 'Haken' sounds more natural.

Die Wohnung ist schön und günstig, aber der Haken ist die laute Straße.
The apartment is nice and affordable, but the catch is the noisy street.
Wir hatten ein Problem mit dem neuen Drucker, aber der Techniker konnte helfen.
We had a problem with the new printer, but the technician was able to help.