little more info would help but from the way you worded it it sounds like its a automatic sounds like a transmission problem to me bring it to a good mechanic
turn the engine off
If would not be bad for a car engine to shift from third to second gear if you are traveling at 30mph.
a noisy second gear in your transmission is that your second gear might have broken gear tooth which is mostly the problem with my cars.either the second gear on the counter gear or the second gear itself.
Motorbikes works same with cars. All we need to do is to start the engine and set the gear to first.
check the exhaust system
The engine will over-rev depending on what size engine you have
Yep, then shift from second to third gear.
you are going too slow for that gear your engine has very little torque at low rpm
Sounds like the front pump in the transmission is tired.
Change the differential gear ratio and/or increase the horse power of the engine.
It doesn't ... it works with the transmission. It's required because the internal combustion engine works best at a fixed speed ... whereas you want the wheel speed to change frequently.
There could be several reasons, such as oil levels, or the engine may be getting old.