Take place sounds more formal and stuffy. It also implies something scheduled or planned and lends itself to describing locations or settings or times. As if... literally taking or claiming a place so that something can occur there. It's like a big deal! "The movie takes place in Medieval France." "The assembly will take place tomorrow at 10 in the auditorium." Happen has a more general, loose usage. You would probably never hear, "I can't believe that took place," but you would hear "I can't believe that happened!" a lot. As for your sentences, I think number 1 sounds fine with both usages, but take place sounds more formal. Number 2 should be something like, "I think in a few more weeks, and then spring will officially begin." sounds most natural. Finally, number 3 sounds perfect as it is.