Thermal Noise in Oscillators
Answer to question 1) 
δ(t) has the dimension of the inverse of time 1/t. Current is charge per time step (I=Q/t). So you have to build the product of charge and the dirac pulse to get impulses of current. 

Answer to question 2) Like Brian already mentioned: A capacitor's capacitance is defined as C=q/V. This can be explained like this: If you put a well defined amount of charge (like 1 Coulomb) into a small capacitor, the cap will reacto to that charge with a huge voltage increase. Vice-versa, if you put the same amount of charge into a very large capacitor, the cap will react with a small voltage increase. That's the linear dependence that C=Q/V stands for.