Taking Glycine and aspartic acid. As both the amino acids are fully protonated at pH 0 ad pH 1, how does the peptide bond form? Is it a two step mechanism where the OH from an alpha carboxyl group bonds with a hydrogen from the alpha amino group of the other amino acid to form water then the H+ from the amino group being released to form a peptide bond?
First, note that the protonated amino group of an amino acid like glycine or aspartic acid has a pKa of about 9.6 - the number for glycine. At pH 1 it is indeed "fully protonated", but more correctly about 1 molecule in 10^(8.6) will have a free amino group at any given point in time (always in flux due to the rapidity of proton transfer). This amino group can react with a carboxyl group (from another amino acid) at the carbonyl carbon to give a tetrahedral intermediate, which in turn can eliminate water to give the peptide bond.
Under equilibrium at pH 0 or 1, the energetics typically favor the reverse of this reaction - hydrolysis of the peptide bonds to give the amino acids - this is what goes on in the stomach when you digest a protein - but it is possible that these conditions could produce a small amount of material from the reverse reaction - the amide-bond forming process.