Learn English – Is it correct? “Fixing this issue will take 2~3 days”

Those sentences are not right.

I don't go jogging often enough that I would buy a pair of dedicated walking shoes.

This does not make sense. It should say: "I do not {do x} often enough that I would {do something related to x}". Going jogging and buying walking shoes are unrelated, so the comparison doesn't work. You could say walking and walking shoes, or jogging and jogging shoes:

I don't go walking often enough that I would buy a pair of dedicated walking shoes.

I don't go jogging often enough that I would buy a pair of dedicated jogging shoes.

Meaning: the person would get little benefit from buying the shoes.

And I'd like to know if this sentence is the same meaning as

I don't go jogging as often as I would buy a pair of dedicated walking shoes.

No they don't mean the same thing; as often as compares how often two things happen, and says which one happens more often. People don't buy walking shoes regularly, and would not talk about how often they would buy shoes compared to going running.

Finally, I just want to double check making another sentence, so please let me know if this sentence is right to use.

I am tired enough that I would need to take a nap right now.

Take "would" out of the sentence, and it works; meaning "I am very tired". Or:

I am tired enough that I would take a nap right now, if I was at home.

"I am tired enough to sleep, but I must not sleep here"

I am tired enough that I would need to take a nap right now, if we were going jogging this afternoon.

"I am tired enough that I would sleep if {...} (but I'm not going to sleep)"

I am as tired as I would need to take a nap right now.

This doesn't have any clear meaning. It's a mix of "as tired as {a comparison with some other level of tiredness}", "as tired as I would need to be {... to fulfil some condition ...}", "I would need to take a nap if { some condition }", "need to take a nap right now".

I "have to" take a leave (or, probably more common leave of absence) is present tense and can indicate "compulsion against the speaker's will."

I "had to" take a leave is past tense and can indicate the same thing.

I "have had to" take a leave is present perfect tense. It follows the normal pattern of forming the present perfect by using have + particle of the verb. In this case the participle of have to is had to. So you get have + had to.

In your sentence, the present perfect means that the leave you were forced to take started in the past and is still in effect at the moment of speaking the sentence.