You can definitely get flexible led strips that are bright enough to do what you want. Many of them run on 12V, which you can easily get from a small power supply, and you could just plug it into an outlet on the counter.
I would put them under the cabinets; it would be easily to hide them there. Many of the strips can be had with adhesive backing, which would just stick under the counter.
To answer your question on the individual lights, the individual LEDs will blend together nicely.
If you look at Amazon, you'll find a lot of solutions, and they are fairly cheap.
Not a complete answer, but here's some more info.
Brightness
While there are ways to calculate this that aren't that complicated, it requires knowing many coefficients that are not readily available, so ends up being an educated guess at best. You can do just as well totaling your current lighting's lumen output, then adjusting for changes in fixture styles. (I consider even a simple LED strip a "fixture" for the sake of this discussion) It's a starting point. If you use proper task lighting, there's a lot of room for variance in total levels.
Uneven light
You do not want to see the actual LED elements, it's really annoying. Besides diffusers, indirect lighting techniques work well, though require more lumens due to losses through reflection.
Positioning
I'm not sure what the issue is with parallel surfaces. The efficiency is mostly inherent in the fixture's ability to direct light where it's needed. The mounting surface has little to do with this, you need to direct the light where it's needed.
White Light
The "whiteness" is measured in degrees Kelvin. The temperature a black body must be heated to emit the same color of light. Standard daylight is often taken to be 6500K, anywhere in the 5000-7000 range is close enough for most tasks. You eyes adapt quickly to small variances. In residential settings, daylight is considered too harsh because we are so accustomed to incandescent lighting, so most home lighting is produced to output more in the 2000-3000K range to mimic the orange incandescent lighting.
Noise
How audible the noise is varies by manufacture and environment, as well as personal temperament. I don't think noise data is readily available. Testing in a noisy store will not yield any useful data, other than if you can hear it from a few feet away it is too loud. Noise can be attenuated by placing it in an enclosure, but keep in mind power supplies generate heat that has to be removed by ventilation, you cannot seal it into a sound proof box.
Try to place power supplies as far from your pillow as possible, trying to sleep is when it will be most bothersome. Placing it lower so there is likely to be more furnishings obstructing a direct path will help, as well as being surrounded by soft materials that do not reflect sound as well.
Best Answer
Your LED strip has one resistor per 3 LEDs, which is the standard for a 12V strip. Each group of 3 LEDs is wired in series, with the resistor setting the appropriate current. All groups of 3 LEDs plus resistors are then wired in parallel.
White LEDs take about 3-3.6V each, so 3 in series mean 9-10.8V, with the resistor taking the rest of the voltage up to 12V.
For a strip, the resistors set the current, so you don't need a constant current driver. You need a standard 12V constant voltage power supply.
3528 LEDs should run at about 20mA each, but remember they are wired in series in groups of 3, so each group of 3 LEDs will use 20mA on your 12V supply. So, 18 LEDs/foot means 120mA/foot, so 1.3 Amps total or 16 Watts.
If you don't fancy getting electrocuted, an enclosed power supply is better than an open frame or naked PCB. I like Meanwell so I'll shill for it. They also make IP68 waterproof models.
If you have a 12V DC wall wart lying around somewhere, that can also work well.
If a group of 3 LEDs fail on the strip, perhaps one LED is dead in that group, or a solder joint cracked.
If one half of the strip fails but the rest works, then the copper trace which carries 12V is broken somewhere around the transition between "works" and "doesn't work."
If all the LEDs are off, or all blinking, then it's either the power supply, wires, or connection between wires and strip.
If they are all wired in series like in some lightbulbs then yes, one single LED failing open will open the whole circuit, but this is not how strips are wired.