Is it possible to customize the java core and heap dump paths for Tomcat?

You can use -Xdump which is documented a fair amount in this article.  I added the settings below to my Tomcat server.startup script and now all Java dumps are being written in /dumps with their associated timestamp/PID appended to the file name.

JAVA_OPTS="$JAVA_OPTS -Xdump:heap:label=/dumps/heapdump.%Y%m%d.%H%M%S.%pid.%seq.phd"
JAVA_OPTS="$JAVA_OPTS -Xdump:java:label=/dumps/javacore.%Y%m%d.%H%M%S.%pid.%seq.txt"
JAVA_OPTS="$JAVA_OPTS -Xdump:system:label=/dumps/core.%Y%m%d.%H%M%S.%pid.%seq.dmp"
JAVA_OPTS="$JAVA_OPTS -Xdump:snap:label=/dumps/Snap.%Y%m%d.%H%M%S.%pid.%seq.trc"
(Optional) JAVA_OPTS="$JAVA_OPTS -Xdump:what"

You can also add the -Xdump:what option above to the Java options to print details in the standard output log when the server starts up.  You'll see this in the log:

Registered dump agents
----------------------
-Xdump:system:
    events=gpf+user+abort+traceassert,
    label=/dumps/core.%Y%m%d.%H%M%S.%pid.%seq.dmp,
    range=1..0,
    priority=999,
    request=serial
----------------------
-Xdump:heap:
    events=gpf+user,
    label=/dumps/heapdump.%Y%m%d.%H%M%S.%pid.%seq.phd,
    range=1..0,
    priority=500,
    request=exclusive+compact+prepwalk,
    opts=PHD
----------------------
-Xdump:heap:
    events=systhrow,
    filter=java/lang/OutOfMemoryError,
    label=/dumps/heapdump.%Y%m%d.%H%M%S.%pid.%seq.phd,
    range=1..4,
    priority=500,
    request=exclusive+compact+prepwalk,
    opts=PHD
----------------------
-Xdump:java:
    events=gpf+user+abort+traceassert,
    label=/dumps/javacore.%Y%m%d.%H%M%S.%pid.%seq.txt,
    range=1..0,
    priority=400,
    request=exclusive+preempt
----------------------
-Xdump:java:
    events=systhrow,
    filter=java/lang/OutOfMemoryError,
    label=/dumps/javacore.%Y%m%d.%H%M%S.%pid.%seq.txt,
   range=1..4,
    priority=400,
    request=exclusive+preempt
----------------------
-Xdump:snap:
    events=gpf+user+abort+traceassert,
    label=/dumps/Snap.%Y%m%d.%H%M%S.%pid.%seq.trc,
    range=1..0,
    priority=300,
    request=serial
----------------------
-Xdump:snap:
    events=systhrow,
    filter=java/lang/OutOfMemoryError,
    label=/dumps/Snap.%Y%m%d.%H%M%S.%pid.%seq.trc,
    range=1..4,
    priority=300,
    request=serial
----------------------

 Alex,