This is weird...
Se the render to Main window, if it isn't already. Then, when you're getting lag, look in the bottom right corner of the window. There will be two speeds represented, the FPS, and the VPS. Are they both at 60 with a 100% next to them?
Yes, they are both at 100% (60) all the time.
If you use the default settings for the controller, aka the keyboard, do you still get lag?
I just tested it more to make sure I'm not going insane. I don't believe there is any noticeable input lag when I use the keyboard. I'm suspicious that the reason for this is that the game somehow interprets the analog stick input differently from the console, causing the input for a dash to register later. When I use the keyboard, the emulator registers a key press as analog stick 100% pressed, so the dash registers almost instantly. Honestly, it can't be more than a few frames of lag so it's not extremely critical. At this point I'm just curious to know what causes it.
At this point, there doesn't seem to be anything wrong with Dolphin or your computer. It could be that something is wrong with the controller.
Is there any other program/game you can use the controller with to test it?
If not, look into a program called Xpadder to map the controller to key strokes.
There is nothing wrong with my controller. Do you know of any way for me to reliably measure how much input lag there is? Maybe some recording program running at a high fps? I know that when playing online, an additional ~80ms ping is added (4-5 frames). I want to compare the amount of lag I get offline to the ping. Thanks for all the help!
Let me reword that: Dolphin may not like your controller.
Did ypu try the Xpadder test?
80 ms is 2.5 frames of latency, not 4-5 in netplay.