Why do i get cold water during the shower every time the pump kicks on

Think of the electricity flowing into your house just like water flowing through a pipe. If you turn on a faucet only halfway then you will probably not notice a drop in pressure elsewhere in the house. But if your washing machine is filling (or your irrigation system kicks in, etc.) then you will see reduced pressure elsewhere in the house.

The electricity in your house works the same way. Air conditioners, refrigerators, water pumps, draw a lot of electricity when they first start up and you may see a drop in "pressure" when any of those items turn on:

Air conditioners, refrigerators, freezers, furnaces, etc. that have large motors require enormous amounts of power when the motor first starts to spin up to speed. Once the motor is spinning, the energy requirement drops way down. This instantaneous need for motor shaft speed is what causes that momentary dimming of your lights.
Reference: http://www.askthebuilder.com/QA_Lights_Dim_When_AC_Comes_On.shtml

As @Tester101 mentioned in the comments, you could look into a service upgrade to your house (which might not be a bad idea) but the problem might also be with your water pump. The pump kicking on/off every 15 seconds may indicate a problem with the pump itself (capacitor or pressure leak) or possibly an incorrectly sized pump for your usage. You probably want to look into a new/repair/upgrade of the water pump.

At 2.5 GPM, your 2-3 minutes of shower time between cycles is using between 5 and 7.5 gallons. If your tank starts at 60psi and 7.5 gallons is enough to bring it down to 40psi, that means you're starting out with 15 gallons of air in your 40 gallon tank, and ending the 3-minute cycle with 22.5 gallons of air. I calculated this using Boyle's law, or P1 * V1 = P2 * V2. And of course these numbers would be different if your shower head is higher-flow and your 3 minutes of showering used more than 7.5 gallons.

I believe to get the most output from your tank, you want it to be nearly empty of water when the pump is set to come on. That means draining and charging it to just below the pump setting, as you did. If you wanted to get a little more cycle time you could probably even go as high as 38 psi. Note that with a bladderless tank, you're going to need to recharge periodically as the air in your tank is gradually absorbed by the water.

For what it's worth I think 2-3 minutes is well within your pump's design cycle time. 15-20 seconds was way too short and by noticing it and fixing it you've probably saved yourself an expensive pump replacement.