Building on Testing Server?

I’m working on redesigning an existing non-wordpress website to a wordpress themed website. I haven’t downloaded anything yet, but have confirmed that the existing web host/server can support wordpress.

Here are my questions:
1) I need to leave the existing website up and running while I develop the new one. Eventually, the new wordpress site will be in the root directory. Do I create a subdirectory on the webserver and download wordpress in to that? What would the url look like for the subdirectory? Is a subdirectory the only and best option?

2) How do I then move the contents of the subdirectory in to the root level and replace the existing content when I’m ready?

3) What issues/problems should I plan on encountering?

Thanks for your help!