What tool to use to map reads (samtools, sequencher,...)?
Samtools is not an aligner; it is used to analyse alignments. FastQC is for analysing the read qualities and average composition. You can get to know the adaptor sequence by running FastQC. 

The next step is adaptor trimming. There are many tools that do that. Trimmomatic is currently popular. You can find the adaptor composition of each read by subtracting the read length post trimming from the read length pre-trimming (this would be same for all reads). 

Then you have to align your trimmed reads to a reference, using an aligner. Again, there are many aligners available; Bowtie, STAR and now a new one called HISAT. STAR and HISAT are faster than Bowtie. 

Since you already have the alignments (aln.bam) to hg19, you do not need to perform alignment. You want to know, to which genes do these reads map to. This information is not available unless you have the genome annotations. The annotations are available in the form of GTF/GFF files. You can obtain the association between genomic locations and the reads using the BEDOPS toolkit as mentioned in this Biostars post.

I don't have much experience with this tool. What I would do is convert the BAM to SAM (using samtools) or to BED (using bedtools:bamtobed) and parse the columns of the SAM/BED file describing the read locations with the GTF file. See this link for the details on the SAM format. You can use any scripting language like awk or perl to parse. 

Samtools does have an option to filter the reads according to regions specified in the BED format but it will not automatically annotate them.

samtools view -hL regions.bed aln.bam &gt; aln_filtered.sam