How do we get the depth for triangle vertices before interpolating inside the triangle?

I have the same question as cheesecake. It'd be great if we can do an example together in class!

Can we do a single set of samples to get both the depth and position, or do they have to be done separately?

Is barycentric interpolation done on demand per pixel or is it precomputed and looked up during rasterization?

If it's possible to calculate using barycentric coordinates, do we need to run through the samples in any particular order, or can we simply calculate depth given the triangle points, sample point, and whether or not the point is in the triangle?

for barycentric coordinates, does the order matter?

Does the floating point precision affects the precision of the depth "d_i"? Doesn't triangles closer to the camera have a higher precision than the ones further away?

Does the order matter when calculating the depth using Baycentric coordinates and also how do we know the viewing angle is it given to us?