The part of summarizeOverlaps() that is based on HTSeq are the 'modes' of overlap: 'Union', 'IntersectionStrict' and 'IntersectionNotEmpty'. The overlaps are computed after records are read from the bam file so sorting by name won't affect the speed of the overlaps.

For paired-end reads, the mate finding is done at the C level. A read is held in the 'to be mated' queue until the mate is found (see 'Pairing Criteria' on ?readGAlignments for criteria). A single pass is made through the data, pairing reads and moving mated pairs off to a different queue. If the file were sorted by name the 'to be mated' queue would be smaller so less to search and yes, likely the mate pairing would be faster. I have not tested this.

FYIs:

• If you are interested in other fragments, not just mated pairs, try out readGAlignmentsList(). This function returns ambiguous and unmated reads as well as the pairs. ?readGAlignementsList for details.
• Sorting by position (not name) is required when you want to extract data subsets with ScanBamParam. In this case you need the sorted bam and index file (bai). Specify positions or fields in the file with ScanBamParam(flag = ..., which = ..., what = ...).

Valerie