Cartopy: Find distance of geodetic/plane carree

Question

I have 4 questions related to cartopy geoetics in various map projections. Here is my code:

import matplotlib.pyplot as plt
import cartopy.crs as ccrs


ax = plt.axes(projection=ccrs.Robinson())

ax.set_global()
ax.stock_img()

ax.coastlines()

plt.plot([-0.08, 132], [51.53, 43.17], color='red',  transform=ccrs.Geodetic())
plt.plot([-0.08, 132], [51.53, 43.17], color='blue', transform=ccrs.PlateCarree())

plt.show()

Two examples of this are below, 1) Robinson projection, 2) Mollweide projection.

Questions

1. Does the red curve always show the great circle route, i.e. the shortest path on any map projection?
2. How can I get the distance of the red geodetic and the blue plane carree? If so, then the distance of the red curve should be the same for any map projection, correct?
3. On my second image, why is the red curve wonky? Is it supposed to be like that?
4. Does the blue curve always show the direct path on the map projection? As if you physically had a map on a piece of paper and drew a path from point A to point B with a ruler. If so, then the distance of the blue curve should be different for different map projections, correct?