How accurate is the driving distance measurement? I’m assuming there’s a small margin for error with changing pin locations. Would this probably even out over the course of the season?

Yes, changing pin locations make a small difference on each shot that tends to even out.
There are also differences due to elevation, temperature, wind, etc. that affect driving distance, but mostly tend to even out (assuming you play on similar courses).
Finally, on straight holes driving distance is slightly underestimated and on dogleg holes it can be slightly overestimated.
But the app does all of the consistently, so the main thing to look for are trends – changes up or down assuming you are playing on the same courses(s).

When inputting the distance off the tee box I typically use the distance listed on the score card. However how does that affect my actual recorded driving distance if the pin placement is in the back which is making the hole longer?

It depends on how precise you want to be. Most players just use the scorecard distance. If you find it easy, it is slightly more accurate to adjust for the distance from the actual tee to the tee marker and the distance of the hole from the center of the green. I’d say on balance it’s not worth the extra effort.

How is the best way to ensure the tee shot distance accurately reflects the club used?

In the ‘standard statistics’ report, refer to the ’75-pct driving distance’. If you use a non-driver on two or three holes per round, these get ‘filtered out’ of the 75-pct value.