Hi Davi, John's answer explains the usage of in and into very well so I thought I would explain the difference between barge and burst. The main meaning you gave is correct but barge is a more violent expression and is an intrusion that is very unwelcome. Burst might be just an unplanned situation. She barged into the confidential meeting to state her opinion. She burst into the room to give him the news. I hope this helps! Let me know if you have any more questions. :)

Both "in" and "into" are OK in these phrases. Often, "in" and "into" are interchangeable -- meaning, either can be used. Sometimes, however, they are not interchangeable. For example, if you say "I am going to leave IN a few minutes," you can only use IN. You cannot use INTO. Similarly, you can say "IN the 20th Century there were two world wars," but you cannot use INTO in this sentence. "The hat is IN the box," but not "the hat is INTO the box." INTO always needs an object. "Burst into the room." But "in" does not. You can just say "He burst in." Technically speaking, I think "in" is an adverb when it follows "burst" and "barge" -- it describes the direction of the bursting or barging. But "into" is a preposition, and it needs to be followed by a place. Another difference is that INTO usually involves a physical movement from outside to inside. IN can be used to mean the same thing ("he went in the house"), but also has the broader meaning -- like to be contained within something (like the hat being in the box, or a war being in the 20th century).