"In order to collect BA Miles you must already be a member of the British Airways Executive Club and will need to provide your membership number at the checkout. If you are not a member you will not be able to collect BA Miles but can still collect Airmiles".
Yes I do it! Not sure about the conversion rate - is it 40 miles for each £2.50 in vouchers but to answer some of your others q's: (i) convert Clubcard vouchers to Airmiles first. You don't get a paper voucher for the miles like some of the other deals, they just go into an airmiles account. If you don't have one already I think that they set one up for you but t takes a couple of weeks. Then the miles just sit in your account until you are ready to use them!
(ii) Don't know if you can use your mum's tesco vouchers - you can't use someone else's Airmiles but if Tesco website lets you credit the vouchers then the Airmiles would be yours! Worth a try...
BA miles and airmiles are different. So 600 BA miles per £2.50 is correct - round trip will be 50,000 miles per person. But just because you have the miles doesn't get you on a particular flight, so I would be amazed if you could go at Christmas. And you still have to pay taxes and charges.
Your mum can also get an exec account and you can combine as a household account, though might have to check terms and conditions.