I was more curious about the #1 "can" aspect, even if the answer was definitively yes I'm not 100% sure I would put it into practice. As was mentioned I would be mortified if I guest saw a reused cork so my wine rack would have yet another element of organization.You ask 2 questions - can you & should you?
I think the answer to #1 is yes, if the cork is still in good condition and has "swelled" back out enough to get good compression for a good seal upon reuse and obviously has no hole through it. The best way to avoid the hole is to use a butlers' thief when taking out the cork the first time.
The answer to question 2 is as everyone has already said.
I would not reuse them for wine but I do use some for various shims on other projects.
lol. Well I would suggest washing the corks my friend...I do wash my undergarments between uses even if I don't buy new undergarments each time I bathe, much to the chagrin of Joe Boxer I'm sure. Love the analogy all the same.It's'like putting dirty underwear after taking a shower