How to obtain impulse response from the differential equation of a system?
It looks like your transfer function is correct, but there's a small mistake in your partial fraction expansion:

$$H(s)=\frac{2}{(s+4)(s+2)}=\frac{1}{s+2}-\frac{1}{s+4}\tag{1}$$

The corresponding impulse response is

$$h(t)=(e^{-2t}-e^{-4t})u(t)\tag{2}$$

The response to $x(t)=te^{-2t}u(t)$ is indeed most easily computed by solving the convolution integral:

$$y(t)=u(t)\int_0^tx(\tau)h(t-\tau)d\tau\tag{3}$$

I leave the exercise of solving $(3)$ up to you, but if I'm not mistaken the result should be

$$y(t)=\frac{1}{4}e^{-2t}\left[2t^2-2t+1-e^{-2t}\right]u(t)\tag{4}$$