right angles (spherical triangle rectangle)
do all intersections between latitudes and longitudes form right angles (90 °) or only the original meridian with the equator forming a right angle (90 °)?
All meridians form right angles with the tangent to each parallel of latitude.
The thing about spherical angles is that they are computed from the triangle sides, which are themselves measured in angular form.
A right-angled spherical triangle can appear anywhere on the sphere. (Looking at a globe, it could be in Mali with the legs pointing in the direction of Greenland and Brazil, for example.)
The easiest one to picture is the one with all three angles being 90°: from the north pole south along the Greenwich meridian to the equator, thence west along the equator to the 90th meridian, thence north along the 90th meridian to the north pole. (You can make one by cutting a melon through its "equator" and then slicing it into four equal parts.)
Beyond the explicit question in the original post, not sure what you are curious about. Napier’s rules and mnemonic relates to solving quantities in spherical trigonometry.
Here is an interesting book on spherical trigonometry you might find of interest: https://www.amazon.com/gp/product/0691148929/ref=as_li_ss_tl?ie=UTF8&tag=theende-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0691148929
“Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry” by Glen Van Brummelen. The Kindle e-book version is only $9.99.
Lost art indeed. I am surprised that the responses did not mention spherical excess...
IMO Computations using a spherical representation of the earth rather than an ellipse reduces its utility in positioning except in navigation or approximations.
No matter you are, from due north/south to due east/west is 90 deg.
Due E/W at a point is the great circle, the prime vertical.