Invision Community supports PNG icons out of the box.
Yes, it should still work.
I don't know the "old" forum icons app so unfortunately I can't answer this 😕
Plugin just replaces default icons like <i class="fa fa-comments"></i> with <i class="classes defined in acp"></i>. It works with FontAwesome 4 or 5 using SVG or Web Fonts depending on Font Awesome version included in your globalTemplate (it's 4 by default).