Hello, I think I can help as I had the same problem! To get to world 4 you must complete world 3 as mini mario which is quite easy and you get a mini-mario power-up right at the end of the castle in world 3. The same thing applies to getting to world 7 in that you must complete the castle in world 5 as mini-mario which is harder as it's more difficult to get a mini-mario power-up! You can only save in the places you said I'm afraid although when you complete the game and restart you will find a 'save' option on your menu screen when you press start. Did you also know that if on the screen where you selcet a file you hold down the L and R buttons and then select a file you can play as Luigi?
Good luck.....I love this game but have done it on all 3 files so I hope Nintendo will bring out a new one soon!