the response of wireless channel to an input
I believe it is worded correctly but might be confusing. The response is a function of time t which includes the response of the impulse which does occur at time \$t=\tau\$

More commonly these are referenced at the same time, but he is just talking general terms.  Typically values prior to \$t=\tau\$ of h are ignored because in a linear/causal system they should be zero when evaluating a channel's impulse response.

The impulse response can be derived mathematically or via measurement.  

Typically in EE applications the system's impulse can be found taking the inverse laplace of the transfer function.  This is a convenient notation to work with because the laplace transform of the impulse function is 1 and time convolution is simply multiplication in laplace space.  Here are some examples from physics.  http://lpsa.swarthmore.edu/Transient/TransInputs/TransImpulse.html