1. First it is important to realize that Hydrogen bonds are weak chemical bonds. When the DNA is in its double stranded helical form the complementary base pairs are basically held in a position for maximum hydrogen bonding to occur between them. When the phosphate backbone is broken, the nucleotides at the sticky ends are no longer physically held in place by the backbone and the weak hydrogen bonds can break (many hydrogen bonds in unison can be strong, but the small amount in the sticky overhangs of 4-10 base pairs is not very significant). The sticky ends of a double stranded DNA that has been cut by restriction enzymes can easily come back together to hydrogen bond too. In fact, when ligase is added to cloning reactions the number of DNA molecules that just re-ligate back together is much greater than those that actually receive the foreign DNA insert. Ligase is needed any time the phosphate backbone is broken.
2. I believe it is possible. For the most part though i think most restriction enzymes act as dimers and recognize palindromic DNA sequences to cut dsDNA.
3. Restriction enzymes and other sequence specific DNA binding proteins recognize the specific sequence through chemical interactions between the DNA and amino acid side chains in the DNA binding motif of the protein. Usually, the protein recognizes certain features in the Major groove of the DNA in which it can distinguish between every type of base pairing! really exciting stuff
There's really a LOT more to all of these things you're inquiring about but you'll learn all of that in a molecular genetics course if you so choose that path!
