LOL that picture made me laugh :)
"badass" is basically equivalent to "cool/awesome" or "tough" depending on the context --in the case of that picture, it would mean "tough" as in, "this guy is so tough he's not even afraid when he sees a bear coming and just goes to sleep" (the picture is a joke making fun of the "play dead" technique to ward off a bear)
dude is just slang for a person. i think most people associate it with males, although where I am from it is used equally for both male and female.