Should I expect decent accuracy?
Should I expect the same outcome if I use RCBS (or any other brand) dies in my LEE press, would it be the same as using all RCBS components??
As far as I understand, charge differences and seating depth play the largest role in determining how each individual round will fire (?)
Yes, you should expect decent accuracy with a Lee Press and Lee dies.
My opinion (other folks have their own opinions) is that as long as your aren't reloading 50 BMG or some other exteme cartridges, and are not a match shooter, the Lee press should be fine. Some presses are much stronger and some guarantee axial alignment better than a Lee, but it is still a reasonable press, although you may with heavy use need to replace some parts at some point.
As to the Dies. My personal preference is "RCBS" dies for revolvers and "Lee" dies for semi-autos. Almost all semi-auto bullets I load have close to a FMJ ogee (sp?). For Revolvers, I go nuts and load semi-wadcutters, wadcutters, rounded/pointed bullets along with hollow points that are crosses of all of the above. The RCBS dies I have come with various bullet seating stems for round, semi-wadcutter, and wad cutter shaped bullets. This gives me much more seating depth control.
The Lee handgun dies bullet seating cup is a one size sort of fits all. If the bullet has a round/pointed nose then it works just fine.
For rifle reloading, if it is a rifle that requires a "crimp," I usually buy the Lee dies with the factory crimp die included. If the rifle cartridge doesn't require a crimp then it is a toss up between RCBS or Lee for me, with a slight preference for the RCBS.
My understanding is similar to yours in the powder charge and then seating depth for a given caliber & bullet are the two prime variables that impact accuracy and in the order you have given them.
For handguns, I have found that on some, degree of crimp can be very important as well. I am sure may folks could argue about run out, primer pocket reaming, trim length, neck pull, primer brand, and a host of other things
Sounds like you have things pretty well figured out and are just looking for some reassurance. You sound like you know what you are doing, so just be careful and have fun.