I would be grateful if you could please help me with the below:
1. Customer from EU is collecting goods from us. What sort of evidence must they supply for us to zero rate? Can we Invoice now at zero rate and if they do not supply the evidence after 3 months issue invoice for VAT only?
2. How should we enter transactions if employees lost receipts and purchases made from unregistered businesses and private individuals? Should these be entered as out of scope or zero rated?
They are risky, in that if you’re unable to obtain the proof of export detailed in that section, you will become liable for the VAT, whether or not you’re then able to recover it from your customer. You could (as you suggest) zero rate now and VAT only invoice later – but (especially if they’re not a regular customer) it could be difficult to get payment from them.
To protect your position, HMRC suggest that you can make providing the evidence a contractual obligation on the customer – and charge them an extra amount equivalent to what the VAT would be in advance, which you can refund to them once you’ve received the evidence.
I would enter these transactions as outside the scope of VAT.