Some answers for the above scenarios:
1. If apt-get says it fails to connect to the repository but nslookup is working:
1.1: Nslookup doesn't tell you if the domain is up, only what ip the domain has, it sends its query to the DNS server, no to the domain.
1.2: Most likely you use a proxy. In my experience, apt-get doesn't honor http_proxy envirorment variable.
You have to write folowing settings in /etc/apt/apt.conf
2. If apt-get says it fails to auhtenticate the packages it downloaded:
2.1: The gpg key from the repository could have an expiration date. I didn't check if it's true for the backtrack repository.
To check it, type to the terminal:
Most likely your system date and time is wrong. Check your system date&time.