If a commercial tool is ok, I can recommend install4j. 1, 2 and 4 are easy to do with it, but about 3 I'm not sure. Edit: You could probably achieve that by installing your app as a "service", which install4j supports for all platforms. Quoting from its Features page:
install4j offers full support for
generating and installing services
(daemons). For services, install4j
generates service executables on
Windows, startup items on Mac OS X and
start/stop scripts on Unix/Linux
platforms.
Edit: Regarding #2, JRE detection: install4 has pretty nice, flexible options for JRE bundling/detecting. You can include a JRE (statically or dynamically, shared or not) or prefer to use one already present, with the possibility to download a JRE from within your installer as a fallback. Check these docs for more: How Installers Find a JRE and JRE Bundles.
(More about my experiences with install4j.)
For other tools, you could check some of the earlier Java installer questions on SO: