Are these chemical reactions with amino acids possible, and if so, what are they called?
Reaction 1

The product is called carboxylic acid anhydride. If you make sure the amino groups don't react (protecting and deprotecting) and get rid of water to prevent back-reactions, this reaction is possible.

Reaction 2

The product could be called a keto-enol tautomer (here, the enol form). The keto form is much more stable, so this product - even if made as an intermediate - would isomerize to the standard peptide bond.