If I get those settings correct or close, does that eliminate the need for exposure compensation?
Yes....if you know what it means to get 'correct' exposure.
Cameras have a built-in 'reflected' light meter and those meters are calibrated to be correct when they are metering a scene that is 'middle grey'. So if your scene is middle grey or if the metered area averages out to that, then it's likely that the camera's recommended settings will be correct (without exposure compensation).
However, if your scene (or the part you meter) is not middle grey (in tone, not color), then the camera's recommended settings will be inaccurate. The farther it is away from middle toned, the more incorrect the recommended exposure will be. So in those cases, the proper exposure will be someone different from the camera's recommended exposure.
The recommended exposure, is the zero on the camera's meter/scale. In the auto or priority modes, the camera keeps the meter at zero and gives you settings to maintain that. So if you are in an auto mode and you are metering on something that isn't middle grey, your exposure will be inaccurate. They way to get a proper exposure in that situation, is to adjust the Exposure Compensation. Which direction you go (positive or negative) and how far you should go, will depend on how bright or dark your scene is.
Exposure compensation isn't a digital setting or something magical, it will just adjust either the shutter speed or aperture...which will alter the exposure.
Now, if you are shooting in manual mode (and you know what you're doing), then you don't need exposure compensation. In manual mode, the camera isn't controlling the settings to keep the meter at zero, so you are free to dial in settings so that the meter is above or below zero...so you don't need to use EC. In fact, with most cameras, there is no EC when you are in manual mode. That being said, Nikon (for some unknown reason) does allow you to set EC while in manual exposure mode. This just shifts the scale that you see, so that if you set the EC to +1, and then adjusted setting to get to zero on the meter...you would actually be at +1.