Anyone know what the missing hose in the first pic is? I found a pic of it attached on another engine in second pic but don't know what hose it is or where the other end goes to.

 

I think it might be the "Vapour Canister Purge Control Valve" tube which goes to the purge solenoid, Part # AJ811695, see this diagram: https://parts.jaguarpalmbeach.com/p/.../AJ811695.html
Yes that is for a MY 2016 F-Type V6 but it's the same engine as yours and the linked page says it also fits the XJ V6, and I just looked at my engine and that area and pipes/tubes look the same as in your pics.

 

I think you're absolutely right; I've had the vapor canister purge solenoid go bad on a different vehicle that caused it to behave the way my XJ is acting. Sweet, thank you so much this was really helpful!