Question 1.
Arrangements where customers collect and export goods are known as “indirect exports”. HMRC cover them in section 6.6 of notice 703: https://www.gov.uk/guidance/vat-on-goods-exported-from-the-uk-notice-703#sect6
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.
Question 2.
I would enter these transactions as outside the scope of VAT.