YTMP3 works on iPhone because it runs in the browser: no app, no install, just Safari. Here is the short version; the full step-by-step iPhone walkthrough (including Files app quirks and Siri Shortcuts) lives on our sister site.
For the detailed guide, including how to handle the Files app, move files, and set playback defaults, read the full iPhone guide on OnlineVideoConverter, which owns device guides for the network.
Yes. Everything happens in Safari; there is nothing to install and nothing to grant permissions to.
Into the Files app, under Downloads by default. From there you can move it wherever you like.