If I were to boil proteins or amino acids in water: At what temperature would the proteins fracture, and then at what temperature would the protein fragments get stuck together again?

I'm not sure, but tests could be applied. For example, amino acids have the property of acids, that is, can turn blue litmus red. You can record the temperature when the litmus changes colour. Then, prepare a biuret test while the amino acids are still boiling, i.e. prepare some aqueous sodium hydroxide mixed with copper (II) sulphate in a test tube. Add the test to the boiling acid and allow to cool. When the mixture turns purple, that means that proteins are present, thus recombining of amino acids.