This is not even a complicated scenario so I share your frustration at getting a simple answer from BA. I2 are an affiliate in the same way KA are - full lounge access rules apply. Here is the relevant page from the lounge eligibility guide - it’s a shame no one you have had contact with at BA has actually opened it!

On threading through your account of what happened on the day (along with the subsequent responses by BA staff when you subsequently sought clarification), I get the distinct impression that this is more about what you perceive to be patronising and rude attitudes than the refusal itself. And on that basis, many people would relate to your unease and sense of annoyance - especially if they had been on the receiving end themselves, so to speak.

In such cases, yes, one could put it down to simple human error, forget about it, and then ‘move on’ as the phrase goes. But there is another view which says that if management are not made aware of staff a) providing incorrect information, and b) handling such situations so poorly in the eyes of a paying customer, then what hope is there of such inconsistencies in service standards ever being recognised and addressed ?

You would be allowed to the CCR if your departing flight to Europe is on the same day as you arrive from New York (and departing Terminal 5). And keep the inbound F boarding pass to show. This is the case for a ET or CE connection. If on a seperate day the answer is a no.

No it doesn’t matter. Just keep the First boarding pass to be sure.

Edit: Sorry didn’t check and looks like CWS beat me to it...