Welcome.
See the documentation.
“Texture Space” affects the generated coordinates, not the UVs.
No. UVs on beveled curves are always (0,0) to (1,1) - there is no way to adjust this on the curves panel - you can only do it through Geometry Nodes or with a mapping node in the Shader Editor (or Python).
For personal projects you are expected to bake everything to export to a game engine. Consider something like Bake Wrangler with a Convert-To-Mesh Python script or with GN if you want to automate it.
Good luck.