Why Butter and Eggs are the Axl Rose and Slash of Baking

Dear Tricia, Why do baking recipes always tell you to have your butter at room temperature? Does that really make a difference? Would butter go bad if you left it out?

Whoa whoa whoa... slow down! You've reached your 3 question limit and I'm not getting paid to write this blog. (Yet.)
While its not life and death, it definitely helps to have your butter at room temperature, mainly because when you add the eggs, you want them to emulsify with the butter, not separate or "break."
Basically butter is almost all fat and some water, and eggs have a pretty high fat content too. (It's a good type of fat, but it's fat.) When you add fat to fat and there's also water hanging out at the fat party, it doesn't really want to mix together well, or emulsify, which is when the water is literally suspended like a little Cirque du Soleil show inside of the fat and somehow it magically remains stable. Ideally, at least. This is hard to do sometimes: that's why people are scared of making Hollandaise sauce from scratch. (Either that, or they would rather just go to a restaurant and have someone else make it. I'm usually in that category.) There are a couple of things to do to emulsify the butter and eggs when you're mixing up your batter.
Temperature is the best way to manipulate butter and eggs into an emulsified, stable mixture. When they're too cold, they absolutely, under no circumstances, will ever come together. Ever. Never ever. Got it? Seriously, imagine cold butter and cold eggs being like Axl Rose and Slash. That magical Guns n Roses reunion will never happen in our lifetimes. So room temperature is a good. That goes for eggs too.
When you've creamed the room temp butter and sugar together and its time to add your room temp eggs, add them one at a time and let each one fully mix itself into the butter before you add more. The speed at which you add fat to more fat is the other key element to making things emulsify. Add it too fast and it will separate immediately. (I could totally make reference to the GnR song "Patience" here, but that's just too easy.) You'll know when this happens because the butter/egg mixture will start to look like it has little curds it it. While its no reason to start over again, it does affect the texture of the final product, so you want to try and avoid the weird little curds of failure by adding things sloooooowly. (And if it does separate, add a little bit of flour.)
And for your third question: no, butter won't go bad if you leave it out of the refrigerator. I'm a fan of leaving butter out all the time, actually, because its easier to spread on toast. Eventually, if you leave it out for like months and years and its August in Florida, then yes, it would eventually go rancid. But fat is a natural preservative and the butter will be safe to use in the time it takes you to use an entire stick, just make sure you keep it covered.