I can confirm the only necessary ports for the ATA to work are your SIP ports as set in your device configuration for the individual lines. (Usually, but not always in the 5060-5080 range) and a UDP range used for the RTP-stream. (This is what actually carries your voice. Also generally defined in your ATA, Linksys often places it in the 16000+ range somewhere, common practice is 10000-20000, which is WAY more than needed, but works great if you're running your own IP-PBX

I see yours is set to 5004 and forwarded from the router as such, 5004-5005 are designated for RTP by IANA thus the Linksys choice here.)
the 13000-13001 ports are specific to the FPL branded Softphones.
Wow that is an incredibly short re-registration period.
Completely un-related, you don't like MWI?
I see you're on a dd-wrt router.
Turn on syslogd under services and under security/firewall turn on the log, I'd hazard a guess at "medium" logging level, for dropped and rejected packets.
Then call yourself until you have the fast-busy issue (or continuous ring) and check the logs. (I don't know if the logs have GUI access, you may need to SSH/telnet into the router, logs are stored at /tmp/var/log/messages
BTW: I'm at work right now so can't check, did you remove the EXIF data from those JPEGs? If not I can recover your SIP Username *evil grin*