1. formamide and/or urea most likely. They both lower the melting point of the ss DNA . This allows the structures to unravel at lower temperatures. Typically you want to reach for melting temperatures around 50° C, and then run the gels at that temperature. RNA is often denatured with harsher agents, like formaldehyd because RNA form stronger structures.

2.ds DNA can exhibit supercooling which is a very elegant method of packaging DNA. This is a huge topic and mathematically derived. It's important is mostly due to it's role in gene expression via the interaction ds DNA has with proteins it wraps around-his tones.

Note that ds supercooled DNA forms two structures; a plectoneme or a toroid, and can also have combinations of these two. It's interesting stuff. Google "DNA supercooling" with an image search and look for a figure that shows the different levels of winding and wrapping:

I'm not very familiar with this source but the figure looks to be on the up-and-up. https://www.google.com/search?client=safari&hl=en&biw=320&bih=460&tbm=isch&sa=1&ei=z_KTU4HAEsqTqgbakYHACA&q=dna+supercoil+hostone&oq=dna+supercoil+hostone&gs_l=mobile-gws-serp.3..30i10.30421.35370.0.35846.13.12.1.0.0.2.489.3028.0j6j3j1j2.12.0....0...1c.1.45.mobile-gws-serp..7.6.975.GZlKkpdiujQ#facrc=_&imgrc=mfy6XWXCf9A4sM%253A%3BMNOv-N10Zf4moM%3Bhttp%253A%252F%252Fclick4biology.info%252Fc4b%252F7%252Fimages%252F7.1%252FDNAfibre.gif%3Bhttp%253A%252F%252Fclick4biology.info%252Fc4b%252F7%252Fpro7.1.htm%3B250%3B250

Good question

1. Formamide and/or urea most likely. They both lower the melting point of the ssDNA. This allows the structures to unravel at lower temperatures. Typically you want to reach for melting temperatures around 50° C, and then run the gels at that temperature. RNA is often denatured with harsher agents, like formaldehyde because RNA form stronger structures.
2. ds DNA can exhibit supercoiling which is a very elegant method of packaging DNA. This is a huge topic and mathematically derived. It's important is mostly due to it's role in gene expression via the interaction dsDNA has with proteins it wraps around-his tones.

Note that ds-supercoiled-DNA forms two structures; a plectoneme or a toroid, and can also have combinations of these two. It's interesting stuff. Google "DNA supercoiling" with an image search and look for a figure that shows the different levels of winding and wrapping:

I'm not very familiar with this source but the figure looks to be on the up-and-up.

Good question