What constitutional right to privacy from private parties? There’s no explicit constitutional right to privacy, and the constitution only binds the government.
Possibly the first amendment, "petition the government for redress of grievances". Privacy violation is not an explicitly enumerated grievance, but neither are most causes for civil litigation.
Also possibly not; it depends on the particulars and the judge.
You also have the right to litigate under common law, which does have a lot to do with the first amendment. Though granted, you are individually unlikely to prevail in that way. Like said, it depends on the particulars and the judge.
I'm not sure where this idea comes from. That clause is treated as the source of the right to access the civil litigation system; this is what "petitioning the government for redress of grievances" means. The right to sue the government itself doesn't meaningfully exist except as the government permits (sovereign immunity), and it was much later that this clause was read (IMO correctly but I'm just some dude) to cover non-litigation activities.