OK, the first thing is that the original sentences are correct. There's nothing wrong with them. If you want to know whether a phrase is common (or correct), you can search for each phrase in quotation marks. For example, search for "take some weeks off", then "take off some weeks". I already know the answer, but I'd like to know how your search goes. ;) The second sentence uses past perfect, which explains why your suggestion is impossible. If you wish to start the sentence with "to...", you need to rewrite the whole sentence instead of changing one part.

He is going to take a few weeks off would be much more common these days but it would be your choice. The sentence sounds incomplete as it stands as we would give a reason for his action.

He is going to take some weeks off. ...ok > off = not at work (adverb) “He is going to take OFF some weeks X Take off > The plane took off. Take off > I took off without telling him because it was an urgent matter. Take off > My singing career took off after I won the contest . Before he became A world champion in boxing last year, he had spent many years practicing very hard. Became is right . He BECAME one LAST YEAR. He won the fight to become the new world Champion . Before he is a world champion in boxing when he won a fight last year , he had been training very hard.