I dread this summer now that I'm running. It's either in the 80's at 6AM with horrible humidity or in the 90's even after the sun has gone down.
I feel your pain, especially now that I'm going to be running every morning, and probably more.
There's not much you can do about it except drink some water before you run, and prepare some way to immediately cool off afterwards. I think it also feels better as you sweat, as sweating itself cools your body down. Just make sure you're hydrated.
And as for bench pressing, aren't you SUPPOSED to (very lightly and briefly, and not let it rest/bounce) touch your chest? Someone was giving you wrong advice, but it's nice that he tried.