It smells like maybe it's running into a mod_rewrite or alias before it's getting to your virtual host definition. Everything you've posted looks correct, but you can try adding a few extra options:

JkWorkersFile  /explicit/path/to/workers.properties
JkLogFile      /var/log/mod_jk.log
JkLogLevel     debug

That first setting will make sure you're using the actual properties file you think you are, maybe something in your multi-tomcat environment is loading the wrong one. You might also check this specific instance's TOMCAT_HOME to ensure it's loading the right server.xml; you can actually pass that as well as a parameter to $TOMCAT_HOME with -config:

cd $TOMCAT_HOME
./bin/startup.sh -config /direct/path/to/server.xml

See where that takes you...

Your solrconfig.xml is located at /app/solr/example/conf:

|-- README.txt
|-- apache-solr-3.4.0.war
|-- bin
|-- conf
|   |-- admin-extra.html
|   |-- elevate.xml
|   |-- mapping-FoldToASCII.txt
|   |-- mapping-ISOLatin1Accent.txt
|   |-- protwords.txt
|   |-- schema.xml
|   |-- scripts.conf
|   |-- solrconfig.xml
|   |-- spellings.txt
|   |-- stopwords.txt
|   |-- stopwords_en.txt
|   |-- synonyms.txt
|   |-- velocity
|   `-- xslt
|-- data
|   |-- index
|   `-- spellchecker
`-- solr.xml

If you set the dataDir to ../solr/data, it means that this directory... doesn't exist. So, change it to the obsolute path /app/solr/example/data and try again.