Is the verse פתחו לי שערי צדק from Psalms 118:19 associated with Hallel? When is typically recited and does it have any connection to ימים הנוראים?
Not only is that verse associated with Hallel: it is recited in and as a part of Hallel. Hallel is recited on various red-letter days, not including the ימים הנוראים (but can, as Psalms and not a special Hallel prayer, be recited any day). (Unless you count Hoshana Raba as one of the ימים הנוראים, in which case, yes, Hallel is recited then.)