Theoretically yes, practically it's not that easy. Line pressure is just one of the components that contribute to the torque capacity of the transmission.
First, it's not the speed of the application of a clutch but the holding strength once the clutch is applied that contributes to strength. In this respect higher line pressure will give higher torque capacity. There are limits though. First you can jack the pressure sky high but if the seals fail because they can't hold the pressure then your up a creek anyway. Next there is a limit to the strength of the clutches and clutch material. There could be lots of pressure but the clutch material fails then your up a creek. If there is lots of pressure and the clutch material holds but you rip the splines out of the clutch disks... With transmissions it's all bout knowing where the weak links of the transmission are and improving the weakest links without running into the next weak link.
Most (not all) of the heat in the transmission comes from the torque converter. As you put more torque through a torque converter it will make more heat. The transmission fluid also works as coolant for other parts of the transmission. If your cooler can't reject the heat fast enough then the components that need cooled will overheat. In this respect the additional cooler will help but it has no direct effect otherwise.