With Showing Off, would a Bot receive any and all "when you flip to" triggers during the process or does it only count the final mode it's flipped to? For example, if
Grimlock - Dinobot Leader is in bot mode and plays Showing Off, could he end in Bot mode with the Bold 3 from his Alt mode?
[Source: WotC]
Yes, you can first flip
Grimlock to alt mode, causing the ability on that mode to trigger, and then flip back to bot mode.
While the alt mode ability triggers when Grimlock is flipped to alt-mode, note that the trigger does not resolve until after Showing Off has completed and Grimlock is back in bot-mode.
For Showing Off, can I flip
Sergeant Springer to one alt mode and use its effect then flip to the other Alt mode and use its effect too? Or must I resolve Showing Off completely first?
[Source: WotC]
Showing Off resolves completely before any abilities that triggered happen.
No, he can not be flipped as part of your ‘designated phases’ per turn NOR battle cards. The only way he can flip at any time is his own start of turn ability.