This might be a broad question. Here is one answer to your question about distance to centromeres that uses operations with BEDOPS tools and UCSC-formatted BED files, which might help you think about your set of questions in more general terms.
First, assuming you are working with human data, generate a file called
centromeres.bed that contains genomic ranges for centromeres:
$ wget -qO- http://hgdownload.cse.ucsc.edu/goldenPath/hg19/database/cytoBand.txt.gz | gunzip -c | grep acen > centromeres.bed
Next, say you have a file containing sequencing reads called
reads.bam. We convert it to a sorted BED file using
$ convert2bed -i bam < reads.bam > reads.bed
Finally, calculate the signed distance between each read and its nearest centromere with
$ closest-features --closest --dist reads.bed centromeres.bed > distances_of_reads_to_closest_centromeres.bed
To now answer your other questions, given your reads (now in sorted BED form), you can think about using UCSC and other data sources to generate BED files that contain telomere regions, GC- and AT-enriched regions of interest, genomic regions that associate with tertiary DNA structure (e.g., ChIP-seq regions or motif binding sites), and regions associated with fragile sites. You could then do set and statistical operations with your reads against these regions using
closest-features and other tools in BEDOPS to help answer these and similar questions.