Current when switching on with inductance. Role of Reactance?
I think to better understand the concept of step response and impedance is useful taking a step back.

The first equation that you write is the step response for a coil with series resistance R and inductance L, when a voltage step function of magnitude U is applied. This step response describes how the current changes in the time domain. There is a particular property of LTI systems that tells you that if you Laplace transform the step response, you obtain the transfer function of the system. 

The impedance it's the extension of the concept of resistance in the frequency domain (or the extension of Ohm's law for time dependent systems), in order to describe resistance, capacitance and inductance with the same quantity. Another way to understand the impedance is to look at it as transfer function between the applied voltage and the output current. If you look at it in this way you'll see that the first representation (step response) and the second representation (impedance as resistance and reactance) are beautifully equivalent, since both are representation of the transfer function of a coil (the step response in the time domain and the impedance in the frequency domain). 

Now, to answer to your question: the step function is calculated using an ideal step voltage meaning that 
\$\ u(t) = 0\ for\ t = 0-\ u(t) = U\ for\ t = 0+ \$ , so the rising time of the edge is 0 (something you would call switching on really really reaaally fast :P). If the voltage step is not ideal. but takes some time to go from 0 to U, that would of course change the shape of I(t) (if you like to think in the form of differential equation the forcing term in the ODE will be different). 

To answer to the second question the R in the first equation is not the total impedance of the coil, since as you correctly stated the impedance is R + jwL. However, that term in the step response (R/L) gives you a very important information about the inductor thats is how fast the current in the inductor will reach the maximum value (something called the time constant).