Phone rings, but no call, and no log entries in Asterisk

1. Well.. if there are failed auth attempts from that number, but you're getting calls from it "from time to time", there's a chance that sometimes, the auth attempt is successful. Try catching the ip address that registers with it by periodically checking asterisk -rx 'sip show peers' for that number.

EDIT (probably a better idea): When the phone rings, don't hang up and look at asterisk -rx 'core show channels' to see what channels are being used for this call. It is possible that the call originates from outside of your network, as the caller id is usually easy to fake.

2. Crank up the log level in logger.conf.

3. Maybe.

Fun fact, 6001 is the number used in Asterisks Hello World Tutorial: https://wiki.asterisk.org/wiki/display/AST/Hello+World

You don't say how large your installation is, but if it's small install the free edition of SecAst, and let it monitor registration and dial attempts. Start by letting it watch the 6001 extension (SIP/6001) and then the PBX in general. Depending on your settings it may be an external hacker trying extensions looking for DISA. A 6 followed by 001 could be the start of a dial-through attempt (6 for DISA, followed by 001 for start of an international call). This may also be a clue that someone is hacking your PBX and mapping your dialplan.