Complementarity of cDNA
It's a good question and has confused me as well. 

A standard definition of cDNA is that it is a double stranded (ds) product of mature mRNA. I suppose it is possible to get a nascent RNA strand copied into cDNA, but I've never heard of that being done.

Because cDNA is dsDNA, the original strand is made from your mRNA as the complementary strand, but then you dispose of the RNA and make the second strand to get a stable, reproducable molecule. The total dsDNA molecule is what we refer to as cDNA (not a single strand of this material).

consider this simplified DNA molecule:

ATG-INTRON-CTCTAG

TAC-INTRON-GAGATC

this could be transcribed into the following mRNA (corresponding to Met-INTRON-Leu-Stop):

AUG-INTRON-CUCUAG

before nuclear export the intron is removed, making mature mRNA:

AUGCUCUAG

you make cDNA from this:

ATGCTCTAG

TACGAGATC

So, you have both a coding and a complementary strand in your cDNA.

As for directionality on the chromosome, genes can be in either direction (meaning that they can use either strand as the coding strand), but remember that the forces that contribute to transcription are just locally determined (assembly of transcription factors, etc) and pay no heed to what direction they're going or strand they are on relative to any other gene. The strands are labeled as positive and negative, but that's just a reference for orientation. A good place to browse the human (or other species) genome is the NIH's map viewer. Click on a chromosome to see the genes mapped to it. Note the column labeled 'O' is for orientation.