Used to be used for all sorts of things. B47s had a bank of them built into the fuselage so the nuclear alert aicraft could get in the air faster.
They are not used anymore because they are wickedly dangerous if the slightest thing goes wrong, and aircraft are generally better. Alert aircraft can start up faster, and engines got more thrust so can simply take off at max power in quite short spaces compared to the early Cold War era.
Fat Albert is the nickname for the Blue Angels transport aircraft. The previous one, a C130T #164763 was fitted with racks for externally mounted JATO bottles, clearly visible here:
For no "reason." Just a party trick when safe to perform, and over the years fewer of them, as bottles became harder to find. They haven't been made since like the 60s, but with no one else using them operationally, lasted a good long while.
That a/c has since been retired and replaced with a former UK service C130J, which I believe does not have these racks.
Agree: someone, somewhere knows all about this bottle and can say who it was issued to when, and likely when it flew. But it's not like aircraft numbers, that won't be public record. Not secret per se but also not public. If you do a hell of a lot of research you can find who to issue a FOIA request to, in order to find out that info.