Welcome to the forum!
For example If I draw a 10 cm line using the grid and then give it an EXTREMELY SLIGHT curve the result is 13 cm. When I have tried longer lines or deeper curves the errors are larger.
I don't understand what you mean. If you have a straight line of 10 cm, and you curve the line without moving the end point, the line naturally becomes longer. And the more of a curve, or the more number of curves, the longer it gets. That's how the Measure Path extension is supposed to work. It measures the actual length of the path, including the curve(s).
If you are looking to acquire the straight line measurement between the 2 end points, then the Measurement tool
is what you need.
For your 2 potential wishlist items:
1 -- It sounds like you essentially need the digital version of a ruler or measuring tape, right? Rather than the distance between 2 marks, you want the distance at the mark, right? And in this case, you do want the curve to be measured? I don't think such a tool currently exists in Inkscape. The Measure Path extension does measure the curve, but it doesn't put the distance at certain points along the path - only from beginning to end.
2 -- For this, you don't want the curve to be measured. You only want the distance between the end points of a line? As far as I understand, this is what the Measurement tool does.