Claiming that AVI is obsolete is like claiming that the plain old telephone line is obsolete. Sure, there's been heaps of advancements made, and you could probably get by without it, but its still in wide use and is still almost as useful as its replacements.
That depends to a great deal on your community. In my area, neither I nor my friends have had POTS for years, and we get by just fine.
Obviously, other communities will differ.
Server farms may be using things like *BSD, but these machines are usually built by hand without automagic package managers.
God no. Server farms (really anything where scalability is a major factor) make good package management all the more important. Now, it might be something home-rolled or a conglomeration of existing package management tools, but something to manage/version software components in a consistent fashion will need to be in place.
The finest eloquence is that which gets things done.