The expiration of the item is independent of the expiration of the coupon or box it came from. So, yes, if you wait until the last moment to activate/use the coupon, you will usually be able to use the item until a later date than if you had used the coupon the instant you got it. I say "usually" because some items expire at a fixed date and time, regardless of when they were obtained.
"Here is a screenshot of how much extra stat the Sengoku Hakase Badge could give you (+8,138,382) "
Yes, it can give you 8m range... if you already have over 100m range.
Or 5k stat when you're at 85k.
And even that is compared with an empty slot, not a Crystal Ventus or Genesis (or Monster Park) badge.
These are not normal numbers that a normal player can expect to gain by replacing their Crystal Ventus badge with a Sengoku Hakase one.
You can expect to gain 4-5% damage output from such a badge. Which is very nice, but it's not as huge as you're trying to make it seem.
(I would very much like for badges with potential to be available in GMS again. I just think you're misleading people by saying the badge by itself is somehow magically worth 5k stat for everybody)