Illumina fastq naming convention changed over the years, once upon a time even the barcode was included in the filename.
Illumina uses underscore to separate the filename fields (which are, in fact, metadata), so using underscores on sample names (as in this case: N2_gDNA_00X) may bring problems if using BaseSpace or other Illumina tools. In reality I don't know, as I never used BaseSpace for data analysis, but at the very least you can not upload a fastq file to BaseSpace if it doesn't follow certain conventions - see here and here.
The current Illumina convention is that, indeed, SX is sample, but the number refers to the sample number at the sample sheet, such as the first sample found at the sample sheet is S1, the second is S2, and so on.
The first 00X in his file names probably means something, but as you pointed out, we really have no way to know what.
Before BaseSpace, the 001 just before .fastq.gz could vary. Files were (well, still are) too big and had to be transferred by ftp. There was (it still exists, but I guess rarely used nowadays) a bcl2fastq parameter for splitting each sample in a fixed number of sequences, so when a file reached that number of sequences, a new file would be created, incrementing this last segment.
are the quality controls produced by http://www.bioinformatics.babraham.ac.uk/projects/fastqc/
your raw reads are only in
N2_gDNA_00X_SX_L00X_R1_001.fastq.gz