Not if they paid by bank transfer! You can only claim a refund through PayPal if you paid through PayPal and used the "goods and services" option.
No chance of a refund using bank transfer.
Not the "business" thing though OP, you're not a business. If they want a warranty or to be covered by the Sale Of Goods Act they need to buy from a business supplying goods, not a private individual selling off their own personal possessions.
Same reason you don't want to accept PayPal "goods and services" when you're selling off your own personal possessions. The law doesn't make you responsible for safe delivery, that's the buyers risk to take, as is choosing to inspect it prior to purchase or not. Your part of the deal ends when you post the accurately advertised item. PayPal "goods and services" makes you responsible for delivering though, so you're unnecessarily giving the buyer more rights than they're legally entitled to.
The rules for PayPal "item not as described" are much more lax than for legal "misrepresentation" too. You've pretty much got to go out of your way to lie about something to get sued for the latter. Whereas failing to mention a small cosmetic scratch that doesn't affect the use of the item might be enough for the buyer to get a refund as "not described" via PayPal.
The risk from handing over too many details is ID fraud. If people have your name, address, bank details, email etc that way too much for a random stranger to have on you.
There's no real risk from doing bank transfer itself as a payment method, all they can do with the info is put money in and it's perfect for if you have to post something.
Just watch out for the money laundering thing PP mentioned above. Report it to the bank/police if that happens to you or they may think you're complicit.