I agree with those who suggested to just go for the wood and put in throw rugs for the sound deadening. If you do go with carpet, definitely check with the manufacturer to see if they have an approved vacuum list. Many of them actually tell you not use Dyson on some products. You are on the right track looking at nylon and smartstrand for a high traffic area. Most pro tests I've seen indicate that Nylon is still top dog as far as durability and not crushing, but Smartstrand is very close. That said, smartstrand is generally easier to clean and more stain resistant. Nylon has the advantage in resisting soiling and if you go with a solution dyed nylon, you can match smartstrand in ease of cleaning. Solution dyed nylon for residential is not the most common yet, though it has been expanding in recent years but I don't think they have it available in nearly the same softness as your soft nylons and smartstrand products. I generally don't look for softness if I am looking at a high traffic area. Other considerations for high traffic - keep to a lowerpile cut or loop carpet. If cut pile, go with a lower pile frieze or textured. Color plays a huge part in hiding soil so look at your mid-tones in particular. Twist should be at least 5.5, 6 preferable, and should have fairly high density.

"This would be the fault of the original designer who might not have cared about a dry basement." It would more likely be the fault of subsequent owners who made chages to the grade or the roof drainage. The water is coming from somewhere. Often your own common sense will show you where. The first steps are yours. Go outside and walk around your house to see if there is anything obvious. Look at the roof line. Are the gutters properly sized, clean and well-maintained. Do the downspouts take the water a couple of feet away from the house? Do the same walk around in the rain, the longer and harder the rain, the better. Watch for damp-looking areas on the surface of the house. When the basement gets damp, is the wind blowing in a particular direction? This may give you a clue where to look. When you are through looking up, go around looking down. Does the ground slope away from the house all the way around the house. If there is paving, does it drain toward or away from the house? Are there plantings (shrubs, trees, etc.) near the house that may allow roots to penetrate the foundation? Do you have close neighbours who may have changed the ground contours since the house was built, directing water toward your property? Do you know the soil and subsoil types? Some soils and subsoils get saturated more easily than others. Are you on city water and drains? The only time our previous house had water in the basement was when the city waste drains couldn't handle the flow and backed up. Assess and correct the outside, before even considering the inside of the basement. You eventually may have to put a membrane around the foundation, but if you can keep surface water directed away from the house, it may not be necessary.

Autumn, looking at your picture I have a question. Was more foam applied after this picture was taken? If not, there should be room in the stud bay to fish in a wire plus add a shallow depth box for 'old work' (it attaches to the drywall, not the stud). It appears (not clear) that the wiring comes from below, and if reachable you could disconnect the wire to that left box and retire it in place (remove the receptacle and patch over). The new outlet would need to be within 24" of the stove, so if that fits the larger studbay under the window, so much the better. The same is true for the right side in that the receptacle needs to be within 24" of the stove rather than 4' from the wall. This might be in that narrow studbay, but that would be up to you. BTW: that 4 foot rule applies to continuous counters. Interruptions by sinks, ranges or cooktops makes the counter discontinuous and measurements start over. The exception is if there is at least 12" BEHIND the interruption (think bar sink).

I'm also in So Cal. I paid about $13,500 for about 700 sq. ft. of engineered wood over slab, no demo, no furniture moving, no steps, no baseboard removal/install. So based on square footage, your quote is more "reasonable" price-wise. But I'll admit I didn't shop based on price as other things are more important to me (quality product, proper prep, skilled installer). I picked out the product I wanted and chose the installer who I felt would give me the best installation, then paid him what he asked for.