Accepted answer
Because temperature enters the rate constant through an exponential, so equal steps in temperature multiply the rate instead of adding to it. Arrhenius puts the rate proportional to exp(−Ea/RT); the working approximation is a doubling per 10 K, which takes 5, 15, 25 and 35 °C to multipliers of 1, 2, 4 and 8. The steps in temperature are equal and the steps in rate are not, and that is the whole of the observation. At 30 °C the same rule gives about 5.7 times the refrigerated rate, and another 10 K would roughly double it again. Met and Trp take up oxygen sixteen daltons at a time, and the oxidised species is more polar, so on a reversed-phase column it elutes ahead of the parent rather than behind it. Ea differs by route, so the ranking of routes changes with temperature too — which is why accelerated data extrapolates badly and why nobody should read a 40 °C study as a fast version of a 5 °C one.
Start with the sequence, because which pathways are available depends on which residues are present.
Hydrolysis cleaves the backbone, most readily at aspartate-proline and aspartate-glycine sequences, and is acid-catalysed. In a dry solid it barely proceeds at all.
Aggregation is physical: peptides unfold at air-liquid interfaces and associate. Shaking maximises that interface, which is why swirling and shaking produce visibly different outcomes on the same vial.
Metal-catalysed oxidation of methionine is documented across peptide and protein formulations and is why chelators appear in some formulations.
Sequence determines which pathways apply, so general statements are general.
Swirl, never shake. Aggregation is a handling problem more than a time problem.