For the first question, the answer is no, unless you specifically design an experimental system to measure such a thing, which is going to be very hard and very costly.
You have much larger objects you can't see with the naked eye. Your entire field of vision is covered by a whole zoo of microorganisms, much larger and much thicker than then DNA molecules that are inside each of them. But you don't see any of them. Stretching a DNA molecule to its full (and truly outstanding) length won't help what-so-ever.
On the contrary, I think you can intuitively realize that by stretching you actually lose visibility rather than increasing it: what happens when you take a piece of paper, which is clearly a visible object, and then slice it very carefully to super thin slices and place them linearly? As the slices get thinner they are more difficult to spot, and so is their shadow.
As for the second one, you're basically asking how strong the bonds that hold the constituents of the DNA strand are. The typical force required to break covalent bonds is in the order of nano-newtons. That's a billionth of the force required to press a key on your keyboard (oh mighty Wolfram Alpha...), which is very very small. To be able to feel the strand, the human touch sensitivity must be in this order of magnitude, otherwise you'd tear it before noticing it's there.
But this is now once more a biological question, which I'd very much like to know the answer to myself! Human senses are actually pretty impressive, but I don't know if they are awesome enough for this task.
EDIT: as for the answer hamilthj gave in Biology.SE, the difference there is that these reasearches are about "resolution", i.e. the ability to notice the existence of very small wrinkles on a surface. But there participants were allowed to touch the surfaces freely and exert as much force as they desired. Here you don't care much for the "resolution", i.e. you don't care about being able to tell exactly where the DNA strand touches your finger, but whether you can feel it or not.
Obviously, if the DNA strand were supernaturally strong it could cut your finger off, but still you wouldn't be able to tell just by toucing were exactly does it contact your hand, and this is probably due to the finite nerve density or something of this sort.
