If you say you're seeing a man about a dog, chances are you're leaving the company you're in for a reason you don't want to give your real reason for doing so. This comes from dog racing, but how it came to be grandpa's favorite euphemism for peeing is anybody's guess. Try asking your grandfather.
Doggerel refers to the crap I spent the majority of my teen years scribbling in my journal - terrible poetry. The online etymology dictionary's best guess is that the word originally referred to something that was only fit for a dog. Which is ironic because dogs hate my poetry.
The term underdog probably comes from dog fighting, referring to the dog that is literally or figuratively on the bottom. Which makes me like the word underdog a hell of a lot less.
Back in high school, I took comparative anatomy as part of a careful plan to avoid having to take physics my senior year. Because if there was one subject I excelled at back in the day, it was avoiding learning things. But I digress as usual. The first critter in the dissection pan in that class was the dogfish shark. One day, I will tell you the tale of what a gutted shark smells like after you've been digging around in it for six weeks, but for now I'll just tell you that I'm proud to say I only puked once. Now the dogfish shark didn't look much like a dog, but it turns out it got its name because it hunts in packs.
By the by, the next critter we gutted was a thing called a mudpuppy. It's the missing link of the salamander family - though it is an amphibian, it lives underwater its whole life - it has lungs, but they don't really do much - the mudpuppy continues to use its gills as its primary oxygen source. They get their name because they're said to bark or yip like a puppy when you pull them out of the water, but I was unable to find a YouTube video confirming this, so it is clearly untrue.
Ever wonder why we call our feet dogs? To my delight I've just learned that this is cockney rhyming slang - in that feet rhymes with dogs' meat.
Info from phrases.org., Etymology Online, and From Hue & Cry to Humble Pie.