This is s a multipart answer due to the complexity of output possibilities:

Measuring within QGIS, distances should be Geodesic, (this will automatically be taking account of the changes in latitude longitude etc along the length of the line for you). In older systems and for long distances On Enroute charts etc the Great Circle distance is commonly reported. Remember that the longer the line the greater of margin for error.

In paper charts of 1:1,000,000 or less you can also use the geodesic distance (in system) if available, Cartographically you could measure distance from the scale bar, however more correct is to read the latitudinal scale and use that to measure distances within that latitudinal range especially north-south lines. Remember all measurements using distance from scalebars and latitudinal scale are approximations across the body of the chart.

Positional information should be calculated using calibrated tools with formulae that are known and verifiable. checking distances on a map plate with a projection and scale can be done as a ready reckoning crosscheck, but should not be relied upon to provide accurate coordinates to be read back into a system.

Calculated positions verified by the procedures designer should pass through the QMS and be verified and pushed back out to the cartographic system as data (AIXM for example).

That's a very good question. Aircraft will use geodetic calculations and most of them if not all are based on Vincenty's algorithms. Aviation data is to be stored in WGS84 so all distances are to be calculated in a geodetic way.

This comes from how you design your procedures it doesn't matter if there are conventional or PBN, so its actually the procedure designer who is setting things here. Let's take the distance between two fixes if you use cartesian and ellipsoidal calculations the difference will be negligible for small distances but as you move further apart the differences will start to be more noticeable.

If the procedure designer uses completely a projected coordinate system (PCS) for design then all the distances and waypoint coordinates he has done are based on cartesian plane (one that is appropriate for the area). There is a common thing I see, IFPD do an exact 6.0 NM on a PCS and then get the lat/lon of the point by converting from the PCS to WGS84, if you do the geodetic calculation then you will not get 6.0 NM. A more correct way but definitely taking more time and longer would be to do all the waypoint calculations on the sphere and then project to PCS to do the design, then at least the waypoint distance will be ok both on paper and the FMS

Here is an online link you may try out https://www.movable-type.co.uk/scripts/latlong-vincenty.html