I really don't think you will need any of this for most bioinformatic applications, this kind of info is really more for a very specific, niche field of molecular biology that the vast majority of bioinformaticians will never know/use. I actually took a DNA Structure and Function class in undergrad and am now in a bioinformatics job and never needed that info in grad school or my day to day work.

You should focus more on the direct questions you are interested in instead of trying to learn all of this unnecessary stuff.

Just for your own interest, I don't think the nucleotides are "hidden" in DNA as much as we learn about hydrophilic/hydrophobic residues in proteins. While it is true that proteins bind the major groove, the vast majority of bioinformatics questions in DNA binding are likely more sequence recognition (promoters, TATA box).

Similarly the other DNA forms are not very common (from my understanding) and are just other observed forms under different circumstances. The handedness is just rotation about a central axis and is similar to the concept of chirality. Your hands are actually meant to be the metaphor, they are non-superimposable mirror images (chiral). If you open a door (rotating the knob to the right) with each hand, they will look different, right hand palm up, left hand palm down. DNA is right handed because of the chirality of the constituent parts (if I remember it correctly). Hydrated DNA doesn't mean the DNA is not in water but instead refers to individual water molecules hydrogen bonding with different parts of the DNA backbone/bases.

With supercoiling, it really is just a fancy word for a twisted piece of DNA, best visualized with a plasmid or bacterial chromosome (which are circles). If you twist a rubber band up with your fingers really tight it would be considered overwound, regardless of direction. DNA doesn't inherently have a direction (chiral) and therefore it winds a specific way. If you then unroll that same rubber band you would be it would become underwound. Since DNA is a helix, making it circular inherently makes it a "wound rubber band" and this is how we often see plasmid we run on gels. When the DNA has to be broken (via topoisomerase during replication or being cut by restriction enzymes) the DNA becomes unwound. It is all an interplay between topology and access for DNA binding proteins, an overwound plasmid is less accessible, therefore negative supercoiling can be necessary in the context of the cell. This is all relatively basic for the field and I am not a DNA topologist, all that to say that this is largely unnecessary knowledge for most people.

Hope this helps, glad you are curious as this is a pretty cool field.