It's actually a very good question. Even up to the Renaissance, we had no real conception of "probability" -- even though there were games of chance that depended on it. The Greeks had notions of impossible/possible/likely, but no way to link the notions to the world.

It was finally in the correspondence of Pascal and Fermat in 1654 that a mathematics for probability started to emerge. With the emergence of the laws of large numbers, this mathematics could be linked to observable real-world occurrence rates. (Ian Hacking's book "The Emergence of Probability" is a standard reference, http://www.jstor.org/stable/2184357?seq=1#page_scan_tab_cont...)

Suppose you were confronted with a stream of random numbers, and no intellectual or mathematical framework for quantifying them. Just coming up with a notion of a frequency curve (i.e., a density), and statistical independence of "unlinked" events would be a huge leap.

I used to TA an undergrad probability class, and in the first weeks, I'd ask the students to quantify some random variable, and they'd sometimes blankly look back and say, "it's just random" -- what we taught them is how you break that down in terms of the familiar quantities (densities, independence, correlations) and operations (multiplying for independence, integrating for expectation, integrating and renormalizing for conditioning).

So to get back to your original question, even a deterministic sequence, like

  0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, ...

could be regarded as random, and given a density and all the other folderol. And a "completely random" sequence of 0/1 equiprobable bits also has a density, etc. And so does everything in between.

Well, we're all familiar with https://xkcd.com/221/ or http://dilbert.com/strip/2001-10-25

The possibility, however remote, exists that you will randomly select a sequence with patterns and order. In cryptography, we define the term 'pseudorandom' in order to discuss a class of functions that look random. Pseudorandom sequences are sequences where, given the first n bytes of the sequence, an adversary cannot guess the n+1th byte with probability significantly higher than 1/2. Basically, it's important to distinguish between blindly selecting a sequence out of all possible sequences, and finding a particular sequence with no discernible internal order.

The idea of what "random" means turns out to not be very simple after all. I think in a typical definition (at least as used for CS?), a "random" sequence of numbers _by definition_ has no predictable pattern to it -- that's what makes it random. Of course, a given sequence of random numbers could _happen_ to have all sorts of patterns in it (and indeed, you could find patterns to post-fit to any sequence), but if you can predict it in advance... it's not "random"? I think?

Yeah, it depends on the distribution you're sampling from. It's possible (often desirable) to introduce a bias when picking a random number. http://www.boost.org/doc/libs/1_42_0/libs/random/index.html

Yes, it's normally one of the first programs most people tackle when learning a new programming language.