Usually the expression is "rode hard and put away wet," but there are variations on it. You are supposed to rub a horse down after a hard ride. A horse can work up a "lather" or a type of sweat, and if you don't curry him before bedding him for the night, he'll be very uncomfortable.
A person who feels like they'd been rode hard and put up wet is completely, bone-crushingly exhausted and weary.
I wasn't just whipped---I was whupped! (Is this the same as saying "I had to work so hard and was totally exhausted. I wasn't just tired---I was very very tired?") And there is a difference. (Does this refer to the difference between whipped and whupped?)
Yes and yes. "Whupped" is simply a matter of pronouncing "whipped" with a distinct southern or "cowboy" accent for emphasis. Feeling "whipped" is feeling exhausted; feeling "whupped" is exhausted to the point of collapse.
wide open, full speed ahead, no holds barred.
All three expressions are different ways of saying the person was working very intensely at full capacity.
(I speak at about 280 words a minute with gusts up to 450.)
He's making a joke about how fast he speaks; the "gusts up to XXX" is a phrase often used by weather broadcasters on television. So this man is saying he talks rapidly as a rule, but sometimes he increases his speed and talks even faster!