Tomcat startup failure with no error output in log that I can find

Question

Sorry, this is long, here is a brief summary: I installed tomcat7 on an Ubuntu 14.04 VPS, it worked (serving the "tomcat was installed" page) until I changed the group owning /etc/tomcat7 to tomcat7 (which was part of a tutorial for setting up the management interface) and called 'sudo service tomcat7 restart'. After doing that, I can't get any installation of tomcat 7 or 8 to serve pages at all on this server instance.

Details: I have no webapps deployed yet, I'm just trying to get the stock ROOT webapp ("tomcat was installed") to run. Exact symptoms - when I (try to) start tomcat, it appears successful. However netstat -a does not show 8080 (http-alt) listening. If I try to reach localhost:8080 (or IP:8080 from a different machine), the connection hangs indefinitely, but I do get 2 established connections on http-alt from the remote machine. There is a question all by itself - if nothing was listening on 8080, how did 2 connections get established when I tried to browse it? There are no errors in catalina.out and most of the other log files are empty. Calls to stop the service (or directly to $CATALINA_HOME/bin/shutdown.sh) fail to connect to port 8005, and report that it seems tomcat isn't running. However, ps shows a java process running tomcat with the same command line as a successful tomcat installation on a different machine. If I kill that process, the machine goes back to rejecting connections on port 8080.

What I've tried:

1. I ran sudo apt-get purge on all tomcat7 related packages, and then reinstalled it.
2. I purged tomcat7 again and got the tomcat8.0.20 binary release. I verified its checksum, inflated it to my home directory, set $CATALINA_HOME and $JAVA_HOME in my environment, and ran $CATALINA_HOME/bin/startup.sh
3. I did step 2 as a newly created user
4. I did step 2 as root
5. I removed the tomcat7 user and group, am not using the same installation location, and have cleared out /Temp/

In case it matters... I am successfully running Jenkins on port 8888. I have tried shutting Jenkins down while working on getting tomcat to run, but it made no difference.

I'm clearly missing something basic here. Is there another stale work directory somewhere that I'm missing? I don't think it is a firewall issue, because even 'curl localhost:8080' hangs indefinitely when the tomcat process is running. I set up tomcat exactly as described in "what I've tried step 2" on my own laptop, and it runs with no trouble, so it seems to me that I've done something to mess up something in my VPS instance, and I am at a loss for what it might be. Sorry that this is so long, I'm so confused on this one that I don't know what may be relevant information or not. Any insight is greatly appreciated.