How can spherical coordinates projected onto a planar coordinate system display data and distances accurately if the earth is a globe? Why should these systems require that? Maps with spherical coordinates are not possible?
This is the first time I've seen you demonstrate the beginnings a modicum of reasoned thinking. Questions like these are the beginnings of knowledge. You're lucky in that others have already answered these questions a long time a go. You can, if you wish, just look up what's already been done, or re-do it if you doubt the veracity of those that came before you. However. that said, I believe your thinking may have gone off the rails several times just in this thread.
1) What they were saying over at gis.utah.org was not that the world is flat. It's that we use flat maps. It was a simple play on words and the technology used for convenience - trust you to take it the wrong way. But of course, if being obviously worng will support your position, by all means, don't let intellectual honesty stand in your way.
2) When they said lat. and long. were useless for calculating distances they meant when reading those from a map projection, and they meant long distances. I think you know that too, so don't go on playing the fool.
3) Lat. and long. are perfectly fine for measuring distances on a sphere. They have to be and if you had any mathematics knowledge at all you would know that. Of course, as a flat earther your use of spherical coordinates is probably minimal, but that's you fault, not the rest of the world's. You are free to learn this stuff you know, or are mathematicians considered in on the conspiracy too?
4) To your question "Maps with spherical coordinates are not possible?", not impossible but certainly impractical. You can get a flat projection map that gives a 10,000:1 magnification, but to accurately do that with spherical coordinates you'd need a globe about 1.3km in diameter. I'd hate to carry that thing around with me, if I could afford one.
5) BTW, the earth is only non-spherical by a small amount - about 1 part in 300, and this non-sphericality is very regular in nature so can easily be accounted for to get exact results using lat and long on our oblate little earth. The last official update to the earth's shape measurements was done in 1984 when the actual geodetic of the earth was measured (using some those non-existent satellites) to 12 digits of accuracy. Applying those corrections lat and long are good for precision in the mm range when applied to the earth's surface.
However, even without those corrections lat and long allowed people to navigate the earth for centuries with amazing and very practical success. How did they accomplish that supposing a flat earth? There must be an explanation, as they did actually do it. Or were James Cook, Ferdinand Magellan and Mr. Polo all shills on the NASA payroll? How they navigated is all a matter of history and you can read up about it. Their methods would not work on a flat earth - can you explain for us pleas? After all, in this thread here you are passing yourself off as a mapping expert. So expert as to debunk all of modern cartography.