Has Anyone Placed Power Conditioner On Its Face with the Power Outlets Facing Up?

 Will that stress cause the transformer to buzz or hum?

No, this has to do with the laminations of the transformer itself as well as the incoming power, now how it rests. Now, if it IS already vibrating and you tweak the cabinet so it's more resonant, that's' another story.

Don’t think it matters.  Keep cords separated, and hear for electrical issues

 mess w cords, rca’s, power cords, CD player, 

I use a Furman line conditioner, and a monster me in 2nd system, monster one is poor quality, tweeter static, etc!

mess w it til u get the sound u want!
replacemcords, rca’s, from CD player, preamp, amps, etc!

 Tinkering is key

If it can survive shipping where it is violently thrown around, it will survive being gently placed on its side. 

I recommend you contact the manufacturer, or in this case Underwood HiFi