Getting different impulse responses by solving with Laplace transform and plugging delta-dirac
The step where you substituted \$x(t)\$ for \$u(t)\$ was incorrect.  It's like saying "\$x(t) = 1\$, so I can substitute \$x(t)\$ for \$1\$ everywhere in my expression".

Observing that \$h(t) \star u(t) = \int_0^t h(\tau) d\tau\$ and then taking the derivative is correct, however.