The short answer is yes it makes a big difference. I would start with using the recommended 0.30 for the heat capacity of the plastic. Once you get a few results which give you an actual value of temperatures (grain/mash tun/strike water temps) and your ending temperature, you can adjust this value to get the number closer to your actual results.
Initial temperature will the the temperature your equipment and grain will be at the start of the mash. You can make standard room temperatures for initially building your profile, but when you brew, use the actual temperatures of the grains and mash tun to give you a better, more accurate target for your strike temperature. Generally, if the mash tun has been in place for a period of time, the temperature will be the same as the ambient temperature of the room; outdoors, it may be a bit different as the temperature may fluctuate depending upon conditions.