Is there a real benefit to removing applications from the iOS multitasking bar? Sometimes, if my iOS device (ipad or ipod touch) is acting sketchy or slow, I will bring up the multitasking bar (double-press the home button) and then remove items from that multitasking bar (do a long touch on one until they jiggle, then press the red delete spot on them.)
I imagine that this is somehow saving memory or freeing up the device somehow.

But is that true?  or are all those multitasking icons simple shortcuts, and deleting them has no real effect?

(bonus side question -- if it does really have no effect, what is a way to free up memory or complexity on the device other than shutting down and restarting?)
In a word, No.

Apple didn't want to hand over the responsibility of application and memory management to the user, and personally I'm glad that they made that decision.

The purpose of the multitasking bar is to allow users to switch between apps and to quit apps in order to open them afresh (in case of crashes or weird behaviour).

iOS doesn't actually allow 3rd party apps to continually run in the background. There are several methods that Apple have provided to developers which allow a sort of pseudo background tasking functionality.

Long and short is, iOS is pretty damn good at managing memory usage and will kill applications in situations where RAM is running extremely low to prevent a total OS crash.

My advice is, let the OS handle it.