Carbon will not soften water, it has no effect on minerals (calcium and magnesium are the primary minerals for GH). If your test kit does not include a test for GH (this is usually a separate test, though some strips have it) you may be able to ascertain the GH of your tap water from your municipal water authority, check their website or call them. You/we need to know the number and their unit of measurement (mg/l, ppm, degrees, whatever). Softening hard water is possible but it is involved and requires diluting the water with "pure" water such as Reverse Osmosis, distilled, or sometimes rainwater (if otherwise safe to use). This can get expensive, and it means having to prepare the water to use at water changes. It is much easier to select fish suited to the parameters of your water. But let's see what they are before worrying about this.
Carbon will remove organics and certain substances (medications for example). It does this by adsorbing (note, adsorb, not absorb) the substance, but this only works up to the point where the carbon is saturated and no further. There is no way to "refresh" carbon, so once it is saturated it will no longer do anything. It will obviously be a bed for bacteria like any filter media is, but nothing more. I agree with other members that it is generally not worth using except when removing medications or similar.
As for rinsing the carbon initially, I would only give it a good swish in a bucket of tap water, then use it. The residue generally does not come off in the filter, but even it it does it is harmless to fish.