Check if startup was successful
There is an MBean that represents the server itself called "jboss.system:type=Server", with an attributed "Started" that takes the value "true" when the server has booted and tried to deploy all applications.
Immediately after this attribute is set to true, tomcat will start it's connectors to let http requests flow in.
2) If you need programmatic check (e.g. from a script), you can try the same thing in (1) using any utility that can get a web page from the command line (e.g. wget http://localhost:8080).
3) There is also the bin/twiddle utility that comes with jboss and can be used to remotely read MBean attribute values. With the following command you can try to read the server "Started" attribute:
twiddle get "jboss.system:type=Server" Started
When twiddle replies with:
You know the server is started.
Now when the server is started, applications can come and go (since they can be hot-deployed / undeployed). I presume the second part of your question is really
Check if deployment was successful
For that you need to look at the MBean called jboss.system:service=MainDeployer. It has operations like "listDeployedModules()" or "listIncompletelyDeployed()". Again, it can be accessed through the jmx-console, or twiddle, or a Java application of yours over the RMIAdapter