Let say I pushed a 100 ISO film to 400, and I always overexpose by 2 stops, so during developing, can I use the normal way to develop the film?
Your question is pretty much semantics. "Pushing" ISO 100 film to ISO 400 means you underexpose by two stops when shooting the film and then "push" the development to compensate. Although not many people actually refer to it as such, when you "push" film in development it is because you "pulled" the exposure when shooting the film. Most people just say they "pushed" the film.
In the case of your example, this would usually mean you put ISO 100 film in your camera, set your meter for ISO 400, and then use the recommended values of the meter set at ISO 400. This would result in the ISO 100 film being underexposed by two stops.
But if, as stated in your question, you increase the recommended exposure of the meter set at ISO 400 by two stops you are then exposing exactly the same as if the meter were set to ISO 100 to begin with! So you haven't really pushed anything when exposing the film. You've shot your film at proper exposure for ISO 100.
The thing one must understand is that changing the ISO setting on a film camera does nothing to affect the sensitivity of the actual film, it only affects the calibration of the camera's light meter to account for what you are telling it is more sensitive or less sensitive film.
If we accidently, or on purpose, err as to exposure, likely the outcome will yield a satisfactory image. This is because the negative / positive method is very forgiving. Think of the negative as just a means to an end. Nobody holds up a negative and says “look at Aunt Sadie, doesn’t she look grand”. We must print the negative to get a positive print. This printing step is tantamount to a retake of the original scene. In other words we get a second chance to expose the original scene, this time on paper. During this second exposure we are able to work our magic and make corrections. This second exposure (printing stage) is the chief ingredient in what we call “latitude”.
We can push the film during the initial developing. As a general rule of thumb the factors to use are: 1.5x developing time doubles the ISO (1 stop). 2 stops = 2.25x and 3 stops = 4.5x the regular time.
That all being said, let’s look again at your question: You wish to push process 100 ISO to 400 ISO -- that’s two f/stops. Further, you said you were going to over-expose. If you over expose 2-stops, this is tantamount to re-rating the 100 ISO to 50 ISO. This will result in an over-exposure. Now you need to compensate by “pulling”. We pull by shorting the developer time. We are talking about a 20% thru 25% reduction in developer time.
About film exposure latitude: We are talking about the range of exposure levels that will yield an acceptable image. For negative materials, exposure latitude is greater in the over-exposed direction. We express these in terms of f/stops. An f/stop is a 2x change, a doubling or halving the normal rating.