"The genome published by the HGP does not represent the sequence of every individual's genome. It is the combined mosaic of a small number of anonymous donors, all of European origin."
I work at a for-profit DNA company which sequences humans for genealogical purposes. The human reference, as we call it, has been through several 'revisions'. My company uses anywhere from "build 36" to "build 38", depending on the specific sequencing product purchased. It's an ongoing effort to upgrade older products to use the newer sequencing version: each revision may change, add, or remove genomic locations of specific named variants. That can wreak havoc, of course, when trying to translate older data to the newer revision. Customers would of course notice right away if their genome data were to change on them! It creates rather a conundrum to solve.
It means we got the sequence of As, Cs, Gs and Ts; but in practice, we don't even achieve that.
The human genome has approximately 3 billion nucleotides in 23 chromosomes. So you would think a "sequenced genome" is 23 strings over the alphabet {A,C,G,T} with a length of 130 million or so each. However, it is only possible to sequence a few hundred nucleotides at a time. So you end up with a jigsaw puzzle of tiny pieces, some containing errors, some missing, some exact duplicates of each other. Putting these back together corectly isn't even possible, doing it approximately is still difficult.
So in practice, a sequenced genome is a sequence of medium size strings, arranged in the mostly correct order, with many small and a few large gaps, so they form a good chunk of the long strings that would really be a genome. And even if all of that was completely correct, we still wouldn't know what most of these nucleotides mean.
The first "draft" of a genome is usually even worse. If you simply dump DNA fragments from a mammal onto a modern day sequencer (Illumina HiSeq), and run software to kind-of solve the jigsaw puzzle, you end up with a million pieces, each about 2000 nucleotides in length, in no particular order. Since most genes by far are longer than that, I'd say it's fraudulent to call something like that a genome, but it's being done all the time.
> Do we ever sample more than one member of a species?
Yes, but not in the same way. Instead of (approximately) solving the impossible jigsaw puzzle again (that's called de-novo assembly), we tend to compare the puzzle pieces from a new individual against the big picture we got from the first genome (that's called mapping). This way, we can see small differences (single substitutitions, small insertions and deletions), which is most of the variation present in a species. What we can't see is large scale variation, like duplicated genes, or rearrangements of large pieces of the genome, or insertions of retroviruses. We know these exist, it's just more expensive to look for them. (All of this changing as sequencing is becoming cheaper.)
Multiple individuals of a species get sequenced when there is interest. Probably tens of thousands of humans have been sequenced by now. For model organisms, the number is much lower. For pathogens, where everything is much easier anyway, the number could easily be higher.
As a practical example, it has been reported that humans and chimpanzees "are 98.6% identical". That's shorthand for "in the detectably homologous parts of their genomes, 98.6% of the nucleotides of chimpanzees and humans are identical". This ignores about 25% of the chimpanzee genome and 20% of the human genome, for which no homologous sequence is known. It also ignores small insertions and deletions, because it's unclear how to define similarity in the presence of those.
There are definitely "similar regions" we have no clue about. Comparisons are definitely not restricted to protein coding regions; we typically compare at least whole genes. That includes introns, which sometimes contain regulatory elements, but most of the time nobody knows what, if anything, they do. The same goes for the untranslated regions of genes (5'- and 3'-UTR), which definitely contain regulatory elements, but for the most part aren't understood. And there are probably homologous intergenic regions, too, and we know nothing about their function.
I don't remember exact numbers about human-mouse comparisons, but I don't need to to know that we end up comparing incomprehensible stuff: We "understand" 1.5% of the genome---that's the protein coding pieces. That's shallow understanding, but at least something. We could pretend that we also understand any piece of the genome that contains a few scattered regulatory elements. Counting very generously, that might be 15% of the genome. The remainder is completely mysterious. In the human-chimpanzee comparison, ~70% of the genomes are comparable. Clearly, we have no idea what most of that stuff does.
Assembling a genome means fitting together all the raw data of short reads into a larger scaffolding.
A few years ago there was an assessment of many of the current methods here:
http://gage.cbcb.umd.edu/index.html
Since then there's been lots of additional work, but looking through those algorithms you can see what some consider "good enough" to call a genome sequenced.