At what point, when connected, do DNA strands become a helix?
In general, a single stranded nucleic acid is helical in the absence of other secondary structure. The base stacking that drives helix formation does occur  between adjacent bases in the same strand. Note, however, that such structures are dynamic and dependent on properties such as temperature and salt concentration. You can see an example of a single-stranded RNA helix in the following crystal structure (1MHK):



Note the blue helix at the bottom of the image is actually a duplex in the crystal.