impulse response - LTI system output

I compute the output of a LTI system, can someone tell me if my answer is right..? and help me with my others questions?

The impulse response is: $h(n) = \left(\frac{1}{2}\right)^nu(n)$ , entry is $x(n)=u(n)-u(n-1)$ in which $u(n)$ is unit sequence.

(1) We know that the outpout of this LTI system is $y(n)=x(n)*h(n)$

(2) If replace we take $y(n)=(u(n)-u(n-1))*h(n)=u(n)h(n)-u(n-1)h(n)$

(3) $u(n)*h(n)=h(n)$ and $u(n-1)*h(n)=h(n-1)$

As a result: $y(n)=h(n)-h(n-1) = \left(\frac{1}{2}\right)^nu(n) - \left(\frac{1}{2}\right)^{n-1}u(n-1)$

My Questions:

1. First of all is this solution right?


2. How we know that the equations (3) stand?


3. Always in these systems in the entry is the unit sequence?