Making sure you're in neutral (manual transmission)?

Generally you want to do it while pushing the clutch. For one, if you are not pushing the clutch you will know if you are in neutral or not, ie if you want to test if you are in neutral but you are in first and decide to take your foot off the clutch before shaking the stick then obviously you will be to late and you will be moving/stalling. Also if you are in fact in neutral and you shake the stick you want to make sure the input shaft of the tranny isn't spinning a mile a minute just in case you push forwards or back a bit by accident and end up grinding the gears, so pushing in the clutch stops the input shaft from spinning.

When you leave the car idling in neutral you dont have your foot on the clutch, but you have to have the emergency brake on.
when you are in neutral you can move the gear selector from left to right. The selector will only move from left to right in nuetral, it will not move in gear, and you should not try and move it without putting the clutch in. Jiggling in neutral will not damage anything.
if you are driving while you are doing this you must hold the brakes and the clutch so you dont roll away, but i wouldnt reccomend sitting in neutral incase you had to move quickly. But no, jiggling the selector in neutral will not harm your car.

Foot on or foot off the clutch, it the shifter wiggles sideways, you are in neutral. Probably best to depress the clutch so you are not only in neutral but the clutch is disengaged.