Foggy is, I believe, taking a break right now, but I will give you my thoughts:
1) The answer to this would depend on the exact terms of your full and final offer and agreement. Many F&F offers are merely a moratorium (break) on regular monthly payments until the IVA is formally concluded. Until that CoC is issued you are still in the IVA and bound by the agreed terms. However, I am confident that a helpful IP would give the required permissions from the IVA perspective.
2) If your IP is happy, as well as the lender, then, yes.
3) You can certainly ask for a letter from your IP confirming that the IVA is all but completed and, at the same time, he agrees to the transaction (if his agreement is still required, depending on the F&F terms). It would then be up to the lender as to whether this is acceptable. I would hope, however, that the CoC arrives before this is even put into motion.
If the CoC arrived this morning, remember that you would still be on the Insolvency Register for up to three months and the IVA will still show on your credit reference agency files until the 6th anniversary (albeit marked completed, in due course). Your lender might still refuse on the grounds of an IVA showing, whether completed or not. If they do a hard search to then refuse, the search on file will discourage other lenders, so, please, do not 'shop around' and only consent to soft searches.
I am not qualified to give advice and can only state my opinions, based on my IVA experiences.