Dinging coming from console every few minutes

Also, the system continues to chime/ding every few minutes while off

XBOX Ding Sound at:

0 Minutes 31 Seconds

2 Minutes 11 Seconds

4 Minutes 11 Seconds

Weird sound at (not sure if that was from the console):

0 Minutes 57 Seconds

I have a suggestion ... call your ISP and ask to exchange your moden I use this moden the motorola and fall straight from xbox live but already had this moden for 6 years after they exchanged my moden never falls from xbox live and you understand how annoying
fall of internet hugs friend

You've had over three different people say it is not as result of the update, yet your persistent on that belief. If you try to be less ignorant and try the suggest solutions above you may actually have it working.

Someone actually had a problem like yours but they were using wireless as their primary, it was suggested that they change their channel (which they did from the default to 11, to 6, and then to 1) which did not work for them, they then asked them to go
wired (and this did not fix the problem either) the person replied saying that they had some success getting online by assigning the XBOX a static IP since it seemed that the IP was renewing every 20 minutes (and the person was getting kicked off every ~10-20
minutes) after trying that they replied saying they would be able to stay online for about ~30-40 minutes and then get kicked offline but would sign back on almost immediately. Their solution was to get the home hub replaced, they replied back a week later
saying their problem was fixed. Now, that's just a suggested solution but it's as close to the problem your facing as I could find on the BT Community forums.