What is the best app to browse, explore and access android files on a Linux machine over wifi?
FTPServer is very good, SwiFTP is another option. I believe they're both in the Market.
I use QuickSSHd. It costs $1.50 US but is worthwhile for me because it gives command line access as well as full root filesystem access. Does require root.
I use WiFi File Explorer. It's for accessing (and moving, etc) files on your phone, nothing else, but it does it wonderfully.