Debian 8 apt-get upgrade fails with Failed to fetch … Connection failed

About your requirements, Iceweasel is the supported Firefox derivative (fork), I'm currently running debian as my desktop OS at work and use iceweasel every day, no problem. Gnome3, I think it'll be available on the next stable release, BTW what release are you running? Squeeze? If so I think (IIRC) Wheezy will have it. And finally, about graphics performance/quality, that depends a lot on your graphics card and its driver, but if you think of it like having transparencies, windows closing with fancy effects and so on, you'll need a moderm desktop or compiz (work with gnome2) which I think its available on stable (wheeze).

I have modified a bit your sources.list for wheeze, do you mind to test it and report back?

deb http://http.us.debian.org/debian/ squeeze contrib non-free main
deb-src http://http.us.debian.org/debian/ squeeze main

deb http://security.debian.org/ squeeze/updates main non-free contrib
deb-src http://http.us.debian.org/debian/ squeeze/updates main

# squeeze-updates, previously known as 'volatile'
deb http://mirror.cse.iitk.ac.in/debian/ squeeze-updates main non-free contrib
deb-src http://mirror.cse.iitk.ac.in/debian/ squeeze-updates main non-free contrib

# 3rd party repositories
deb http://packages.dotdeb.org squeeze all
deb-src http://packages.dotdeb.org squeeze all
deb http://www.deb-multimedia.org squeeze main non-free

If you happen to be using wheezy or sid you'll have to change all squeeze ocurrences for the one you're using.

Please, backup your sources.list before replacing it, then test it as follows:

Refresh caches

# apt-get update

Search package

# apt-cache vlc

Install package

# apt-get install <package_name>

If find trouble, please report back with output for those three commands, or at least for the first and last.

Also, if you are already using Wheezy or sid, forget what I said about Gnome3 not being available, it should be there but wheezy is yet to be release and sid is always the development branch.

First of all, despite the last line start with E: (which indicates an error), apt didn't fail entirely; it's downloaded most of the updated package lists, it only skipped those for Opera and the Google Talk plugin. So apt upgrade should still offer to upgrade all the other packages.

The warnings give you some indication of what went wrong:

W: gpgv:/var/lib/apt/lists/dl.google.com_linux_talkplugin_deb_dists_stable_Release.gpg: The repository is insufficiently signed by key 4CCA1EAF950CEE4AB83976DCA040830F7FAC5991 (weak digest)
W: gpgv:/var/lib/apt/lists/deb.opera.com_opera_dists_stable_InRelease: The repository is insufficiently signed by key 419D0ACF314E8E993F7F92E563F7D4AFF6D61D45 (weak digest)
W: Failed to fetch http://dl.google.com/linux/talkplugin/deb/dists/stable/Release  No Hash entry in Release file /var/lib/apt/lists/dl.google.com_linux_talkplugin_deb_dists_stable_Release, which is considered strong enough for security purposes

The first two mean that the repository descriptors were signed with an old digest algorithm, which apt now complains about. The third is caused by apt's recent switch to SHA-256 or SHA-512 hashes only; the Talk plugin repository only provides MD5 and SHA-1 hashes, which are now ignored by apt.

To fix this, you can either remove the repositories for the time being, or wait for Opera and Google to fix them...