For piston/prop aircraft requesting intersection departure, is it pilot discretion for the minimum amount of runway length required? Is there a takeoff calculator in conjunction with calculations?
For the first part of your question, yes. It is always the pilot's discretion to either request, or to refuse/accept an intersection departure if offered by the tower.
To the second part, yes as well, but "takeoff calculator' is a little unclear. All piston prop aircraft, (really all certified aircraft) have performance charts in the Pilot's Operating Handbook, (POH). Pilots are trained to use these to determine ground roll distance, obstacle clearance, and other takeoff related performance numbers based on weight of the aircraft, density altitude, and other parameters. (more sophisticated aircraft have automated tools available to help with performance calculations.)
So, the decision in the first part of the question should be based on the knowledge derived from preflight planning described in the second half of my answer.
All general aviation pilots are required to know the takeoff and landing distances needed during the operation of a flight and the runway length available, per §91.103. So to comply with safety and legal requirements, a pilot had better well know how much runway is available in an intersection departure prior to attempting it. This not only includes the distance required for a ground roll, but that needed to clear obstacles at the departure end of the runway, and offer the required accelerate-stop distance, if operating a multi-engine airplane.
The decision to accept an intersection departure is ultimately at the discretion of the pilot and the pilot is fully in the right to refuse an intersection departure and request a full length departure if they don’t feel comfortable with doing so.
Because people do have their right to privacy, few comments other than these are offered. Nevertheless, let's think about this for a moment. An expedient action can have unintended fatal consequences. The first and last paragraphs of the introductory summary of the NTSB accident report read as follows -
The pilot and passenger departed on a local flight in the amateur-built experimental airplane using an intersection departure from the runway. Several frames of a surveillance video that captured the airplane during the initial climb from the runway showed a thin white trail of vapor/smoke behind the airplane. GPS data recovered from the airplane indicated that the airplane climbed to about 100 ft above ground level, leveled off, turned left at the end of the runway, and entered a descent. Shortly thereafter, the airplane collided with parked vehicles about 0.25 miles from the departure end of runway. Examination of the accident site indicated that the airplane impacted in a near-vertical attitude, consistent with an aerodynamic stall.
The circumstances of the accident are consistent with the pilot failing to maintain sufficient airspeed following a loss of engine power during takeoff, resulting in the airplane's wing exceeding its critical angle-of-attack and a subsequent aerodynamic stall. Instead of using the full runway length of 6,179 ft, the pilot elected an intersection takeoff with about 2,570 ft of available runway. Calculations showed that, had the pilot used the entire runway for takeoff, sufficient runway for a landing following the loss of engine power would likely have been available.