The peak temperature will be within the shotshell hull and will occur during the time of peak pressure. The pressure will peak within the first couple of inches of shot travel; usually early within this period. Shot acceleration is directly tied to pressure, so the peak acceleration will be very early in the shot travel. Peak velocity, however, will occur at the muzzle as the shot continues to accelerate all the way down the barrel; at least for useful barrel lengths. The acceleration is down to around 20-40 additional fps in the last few inches. If you were to make a long enough barrel, the friction forces and air back pressure would eventually equal the force of the powder gs pressure and the shot would stop accelerating; longer and the shot would actually slow down.
Speed of sound in a gas is typically related to the square root of the temperature; higher temperature equal higher speed of sound. So, it is non-linear.
