Does a small change in DNA, like a missing nucleotide always affect the protein?

yes because they way a protein is made has to do with the order of nucleotides in its corresponding dna strand. One amino acid corresponds to a group of three nucleotides (codon) so if a nucleotide is missing it shifts the way the dna is read (meaning the codons are no longer the same) and the amino acids are different, which results in a completely different chain of amino acids. However, since more than one codon can code for the same amino acid, if one nucleotide is simply replaced by another but the amino acid that is coded for stays the same then the protein is not changed.