I didn't watch the video, but I've done DNA extractions in school myself so ...
1 -- Yes, very likely. The helical structure is preserved, its really part of the secondary structure of the molecule. Research a bit more into why the helical structure happens and you can see why this is so.
2 -- Since it is visible to the naked eye, its not a single molecular strand, but instead millions stacked on top of each other. Possible with some other chemical impurities. But the gooey white strands do indicate DNA molecules stacked on top of each other. It tends to associate with itself and water in this way, because oif its chemical properties. Which you can read about.
3-- Well ... how will you get it without a laboratory? And what will you study about it without a laboratory? Kinda a moot point, isn't it? However, if you repeat the steps, using this as an input source, to remove yet more cellular components.
FWIW, I reviewed the comments on the movie on Youtube, and many people are asking questions like yours -- can you prove this is DNA, and only DNA, with a light microscope or something? And no, you can't.