No filter removes all contaminants. Carbon is great for improving the taste. The white ones remove large sediment. Activated alumina removes some fluoride among other things. Distilled water will be free of most things but volatile chemicals will make it through. UV light and micropores (ceramic and "nano tubes") will remove most virus bacteria protozoa and cysts. Uv light will kill most if not all of those little nasties if the flow is right.
So the first step would be to go get your unfiltered water tested. Then see what's in it that concerns you then figure out what you need to remove those contaminants. Your filter might not be designed to remove the stuff you have a problem with.
Water softeners do use water to recharge, but it shouldn't be that much. With a new filter and all electronic controls working, most systems should send no more than 25 gallons through during a recharge cycle.
The theory is that the filter media (special polymer beads, often stored in what looks like a high-pressure air tank) is "charged" with a high amount of sodium ions from the rock salt you load into the bin. Hard water containing calcium and magnesium carbonate, which are only slightly soluble and cause lime scale, comes into the filter from the house supply, and the sodium replaces the calcium, forming highly water-soluble sodium carbonate in the water supply, which won't cause scaling, leaving the calcium and magnesium ions in the filter. To "recharge", the softener fills the salt bin with supply water producing a brine, which is then flushed through the filter media. The sodium replaces the calcium and magnesium in the filter, which then hitch a ride down the drain as soluble chlorides.
The amount of water needed to produce the recharging brine varies by manufacturer and age of the filter media; this ion exchange does take its toll, as the charged ions are corrosive (a surplus of positive ions in a water solution makes the solution acidic). The softener also relies on several detectors to determine system status, that are exposed to hundreds of thousands of gallons of water over their rated life, much of it salty and highly corrosive. Over time, all these components wear out, with more or less detrimental effects on the system's overall efficiency.
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Water is the amount of water coming into your house. Sewer is (usually) the portion of it that you use inside the house and that goes out through your drains, toilets, etc. See #2 below. Sewer charges may also include a portion for storm water if your downspouts feed into a municipal sewer system.
During the summer, it's expected that some of the water will be used for watering your yard, so your sewer usage will be less than your water usage. During winter, the two should be very similar.
In general, water in bathroom fixtures should be safe to drink, boiled or not; there's only one water main coming to your house, after all. You may have additional filters on the water lines going to the kitchen faucets, which would make that water taste better (by removing chlorine, say).