DNA has two sets of bases. Why is this the case? If one gets mutated, does the other as well?
One could argue that any informative polymer must have more than two different monomers in its primary structure in orden to contain biological information. One could make an analogy to having a language in which any word contains two letters. Any word contained on such languague, like aaaabbbbb or ababababa for example, would essencially be devoid of information, as we can predict the the sequence of letters. So it is clear that if DNA was to be the molecule carrying the genetic information of a particular cell, then its primary structure must contain more than two different monomers. Now, asnwering why exactly does it use 4 different nucleotides is quite hard.
When you use an even number of nucleotides, all the nitrogen bases can have its complementary base, which allows for the formation of two or three stabilizing hydrogen bonds, in sort of a one-to-one relationship (cytosine to guanine and adenine to thymine). These hydrogens bonds can form because the functional groups of the complementary bases are arranged in space in such a way that allows the formation of these stabilizing eletrostatic interactions. If DNA used any odd number of nucleotides, like $3$ or $5$, this one-to-one relationship would not be possible, since you would always have an extra base without a complementary partner. If DNA uses $3$ different nucleotides, let's say, thymine, adenine and cytosine, we can see that in this example, cytosine lacks a complementary base to bind to. The same can be applied to an usage of $5$ different bases, and in general, to an usage of $2n + 1$ bases. Then, the extra base must bind to a non-complementary base and form a much weaker interaction with it, as the functional groups of the bases will not complement in space well enough, which will decrease the stability of the molecule. So, we can see that, any DNA molecule that used an odd number of nucleotides would be far more unstable than any other that used an even number of nucleotides.
Now, in terms of an even number of nucleotides, from above, we can see that DNA could not use $2$ different nucleotides, as it will hardly contain any biological information. The usage of an even number of nucleotides that are greater than $4$, such as $6$ or $8$, seems to be quite unfavorable, as this would imply that we would have to recognize hundreds of different codons. If we pick greater even numbers than these two, the situation only becomes worse, as the number of codons would exponentially grow ($n^3$) into the thousands. So it seems that using $4$ different nucleotides is, by far, the best pathway to take in terms of stability and practicality.
As for the second question, no, a gene mutation on a particular base will not necessarily alter the complementary base, but it will affect the number of hydrogen bonds the pair can form, and therefore, the stability of the molecule.
NOTE: Credits to Bryan Krause for his insight.