I lived in Australia back in 2001. Will my tax number and Medicare number still be valid, or do I apply again?

Your tax file number is for life, so it will still be valid. If you've forgotten it, phone the ATO on 13 28 61 for help.

Your Medicare card will have expired so you'll need to visit a Medicare branch with the required ID to get a new one. More information here: http://www.medicareaustralia.gov.au/public/update/faq.jsp#N1006E

The above is true. I had to apply for medicare again but the TFN was the same when I returned 20 yrs later