It sounds like you want to do no-boil with extract. I've never tried this before, but I think it could work, if you pay extra special attention to sanitation and yeast health, and don't take any shortcuts.
Like you were thinking, you'll need to pre-treat all of your water with Campden. The right dose I believe is 1-2 tablets per gallon, crushed, and you'll want to add it to all of your water in bulk about 2-3 days ahead of time. Otherwise if you don't wait for days, the Campden will kill your beer yeast in addition to everything else.
Softener water is okay, but like a10t2 said, it might end up tasting salty from the sodium. You could counter this by adding RO or distilled water, perhaps at a 50/50 ratio. I would not use your raw well water, which I would guess is relatively hard and nasty. Soft water is a must for extract beers.
Sanitation of your fermenter and stirring spoon and all equipment will be of the utmost importance. Use a good quality sanitizer such as StarSan to sanitize everything in accordance with manufacturer's instructions.
Yeast is another factor. You'll need to pitch sufficient healthy yeast. You can use the calculator at mrmalty.com to ensure you're using enough yeast. If you use liquid yeast, a yeast starter is absolutely essential -- not optional. If you use dry yeast, you'll need to pitch enough packs of it per the calculator. Don't skimp on the yeast.
Let us know how it turns out. I'm curious to learn more about no-boil brewing, if that is indeed what you are trying.
I would also encourage you to reconsider not wanting to boil. There are lots of advantages of heating and boiling, including hop bitterness, ability to steep specialty grains or mini-mash, beer clarity... the list goes on.
Also consider whether you might be a good candidate to boil half your recipe (e.g., just 3-4 gallons) then top up with another 3 gallons clean sanitary water after the boil. Then you receive most of the advantages of both methods.