insert size calculation and with this plotting of the nucleosomal ladder (see original ATAC-seq paper) which is a helpful quality control metric. If it is not present (and there are published datasets that fail on this metric) I would not use the data.
Knowledge of fragment length has been used to perform "nucleosomal positioning" analysis (see Buenrostro 2013 and Schep 2015 "NucleoATAC").
PE gives more information on the cutting sites. As in ATAC-seq, the start and end of a fragment mark the insertion point of the transposase. Given PE sequencing in contrast to SE you will also get the end and not only the start of a fragment and by this additional information, e.g. when creating a count matrix for differential analysis. I always count fragment ends when creating count matrices so PE will give you double the information compared to SE.
PE is nice to have but I think not absolutely necessary especially if your goal is only to call peaks or standard differential analysis. Probably depends on money etc, PE is better (and on most Illumina platforms not so much more expansive than SE, especially since the Novaseq becomes more and more prominent), but SE in many situations will also do. If you need as much information as possible, e.g. for footprinting applications, definitely do PE.
While it is certainly possible to do single end sequencing on CUT&RUN
libraries, generally we do not recommend it becuase you will lose the
fragment size information. CUT&RUN realize on MNase to processively
digest the DNA down to the minimal protein protected region, by doing
paired-end sequening on CUT&RUN libraries it is possible to obtain a
footprint of the proteins bound to a particular region of the
chromatin during digestion. In addition, paired-end sequencing
provides more information per read allowing you to reduce the
sequencing depth required to obtain high quality signal for TFs,
Histone Marks, and chromatin modifiers.