Is $[0,1]$ a topological group? Can one endow the unit interval $[0,1]$ with a group operation to make it a topological group under its natural Euclidean topology?

No. A topological group is homogeneous, and $[0,1]$ is not, since it has the two endpoints. (An open neighborhood of one of the endpoints, like $[0,1/2)$, is not homeomorphic to any open neighborhood of an interior point via a homeomorphism mapping $0$ to the interior point.)