Hi,
Two things.
1) IS the ADSL modem routing as well.
You should not have two DHCP servers on the same network.
If it is not in some form of Bridge Mode does it have the ports for iChat Open ?
2) In your Linksys in the Security tab if you have one does it list a Firewall ?
If so turn this Off as it has Features that interfere with iChat.
Also uncheck the Box about Blocking Anonymous Requests on the WAN port. This is Ping Blocking and iChat sends a Ping.
From what you post, the ADSL Modem is set up for iChat probably with UPnP as Screen Sharing works.
Ideally the ADSL Modem should be set to do Static Routing to just the Router.
The router can then do DHCP to any computers.
This pic of the top of a Linksys front page (SET UP) may provide clues about setting static Routing. (Set to Static after the Modem has been set to not do DHCP and then fill in the IP you want it to have, the Subnet Mask, Modem's IP (Gateway) and the DNS Server info) You will lose contact with the Internet whilst you do this.
The Double NAT may still be a problem but will be better than Double NAT and two DHCP servers.
If there are two DHCP servers iChat will see two connections to the Internet.
The Ping in iChat is to confirm where the Visible Invite went to is the same place the Connection is trying to connect to.
To see the two DHCP issue graphically
see here in the slowed down .gif
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9:03 PM Saturday; December 5, 2009
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"