Yes a private investigator can lie but the lies won't hold up in court.
If a private investigator goes to court with you or the person who hired them then the private investigator has to agree to tell the truth.
if the private investigator lies in court then they can be jailed or perjury.
So most legit private investigators won't lie but it's possible that some private investigators may lie but that is rare.
Private investigators are allowed to take photos of someone as long as the private investigator is taking the photos from a public property such as on a road, sidewalk etc.
The private investigator cannot go up to someone's window and take photos through the persons window but they can take photos of the outside of the house or take photos of you when you're outside your house.
Just like anyone else can take photos of the outside of your house or take photos of you out in public the private investigator can as well.
Most private investigators are working for a company or for themselves and are not law enforcement.
The non lawn enforcement private investigators have no authority over you and cannot make arrests and have to follow the law like every other citizen.
Sometimes private investigators are hired by law enforcement to carry out investigations and those private investigators could actually be law enforcement and have authority.
However the private investigators that work for the public and are hired by the public to follow someone and investigate things or people are just regular people that are non law enforcement.
Those private investigators have no authority to arrest someone or any other type of authority and must follow the same rules and laws that other people do.
Most private investigators will indeed follow the laws rules and regulations so they do not break the law and get into trouble.