Please help, I can’t catch the reasons of the following sentences, any explanation and help would be much appreciated.?

Some of these sentences are probably acceptable in casual American English. They would not be acceptable in British English.

Some are incorrect and unacceptable in American or British English.

Let's treat the American versions and a contraction of a possible British sentence.
Let's start with a sentence that is closest to making sense, "I think I'll go put my jacket on". The English version would be "I think I'll go and put my jacket on". It is not the same as "I think I'll put my jacket on". If you have got the jacket with you, you would say "I think I'll put my jacket on". If it is somewhere else, you need to go to it before you can put it on so you might say "I think I'll go and put my jacket on".

To 'put your jacket on' is an event. To 'wear your jacket' is a continuing process. You first have to put it on. You can go to do an event (put my jacket on, brush my teeth, or whatever) but you can't go to an ongoing process (wear my jacket, be fifteen years old).

Your second examples are both wrong. "Do you prefer to go wear underwear or go commando?" does not make sense in English. The expression 'to go commando' is a slang expression meaning to not wear underwear. The sentence should say "Do you prefer to wear underwear or go commando".

The next sentence is even more wrong," Is it professional to put my cell phone on a belt clip case go wear it at work?". It does not make sense in English. Have you copied it correctly? Are there words missing? I cannot work out what the sentence is trying to say. The first part if fine but the meaningless part is "case go wear it at work." Whatever it is supposed to say, the expression 'go wear' would still be wrong. The verb 'go' is inappropriate in that context.

all your sentences are total gibberish. u should NOT learn english from this website. NOBODY talks like this. NOBODY. nor in britain neither in usa.
WEAR is a normal verb, u don't need to use another verb "go" together with it. do u prefer to WEAR underwear or go commando? and the second sentence is a total gibberish, what does it even mean? anyway, nobody says "go wear" in any country. u may say "go ahead and wear an underwear" but this is something else