After 4th ring, there is a loud buzzing noise.

SOURCE: Answering machine doesn't pick-up reliably

I found in the manual that if the machine doesn't answer after the correct number of rings, then it may be the ringing system used by the phone company. But I have also heard of people with VOIP (such as Vonage) having problems with this as well.

http://telephones.att.com/docMgt/public/art/21714/1738_english_manualpart2.pdf

SOURCE: HELP

buy a new answering machine, this one is toast

SOURCE: #1739 answering machine hooked up but keeps asking for "password"

The unit appears to be stuck in "message retrieval mode) and is asking for your password to give you access when you call in(remote answering). You need to return the unit to it's factory default settings. If you don't still have the instructions you can get them here:http://telephones.att.com/docMgt/public/art/309494/1739_manual.pt2.pdf. Hope this helps!

SOURCE: disabled phone line

Sounds like the answering machine is picking up the telephone line and not releasing it.

Hence the busy signal, cause the answering machine has the line off hook. It's like leaving your telephone off hook, the next caller will get a busy signal...

The answering machine might be bad... have you noticed that you were talking on the phone and hung up then the next call would be the busy signal? Maybe test this out.
Do you have the correct voltage to the answering machine, too much power or too little power can do funny things to telephone equipement.