The fact that many branches of mathematics have no application in the world demonstrates that mathematics is not derived from anything in the world, or validated by the world. We can have mathematical truths irrespective of there being an application. That some areas of maths have application is no mystery. For as long as people have dreamed up mathematical ideas, they have also looked for ways to make practical use of them. Some work, some don't.
None work perfectly.
Possibly, but in the absence of humans there still doesn't seem to be anything that could be true. That doesn't seem to follow unless you take a strongly Platonist line. But I'm still not at all clear what you are claiming is necessary and what in the world makes it necessary.