Finding the nearest gene to a specified gene region
You're looking for bedtools closest, which thankfully defaults both considering overlapping genes and outputting all hits in case of ties.

  bedtools closest -a foo.txt -b genes.bed | awk '{foo = sprintf("%s:%i-%i", $6, $7, $8); OFS="\t"; print $1, $2, $3, $4, $5, foo}'


You'll need to strip the header and ensure the file is sorted. I've piped the output to awk to reformat things. The output then is something like:

chr1    24613189    24613641    0   3   chr1:24613188-24613971
chr1    24613971    24614812    0   4   chr1:24613973-24614651
chr1    35788729    35789387    -5  0   chr1:35806973-35810462
chr1    93151586    93160225    0   -2  chr1:93151348-93160948
chr1    93151586    93160225    0   -2  chr1:93151582-93160870
chr1    107511454   107525597   -2  0   chr1:107511422-107525598


Those results are for the Gencode M15 release (the BED file I used has transcripts rather than genes, so you'll get a bit different coordinates). As an aside, the gene name would normally be output as well, just add column 9 to the output.