Door 4 is substantially faster than other doors. You can pretty easily get door 4 by staring at the 4 and following it with your eyes. There are also several eggshots that can be consistently performed to get it as well, as shown below.
|6-8 Gate 4 Egg|
The fastest baby bowser strat is to bounce on his head, then immediately ground pound.
|6-8 Baby Bowser Fight|
To spawn eggs on big bowser, do not scroll the screen. Wait, and they will come to you. Aiming eggs at bowser is very difficult, as his hitbox is very small, and constantly moving. However, you must be firing eggs near the middle of his body in order to hit him (i.e. Yoshi cannot be too far left or right).