I've been hitting an issue intermittently and I'm looking for help in troubleshooting it.  If the best thing would be to file a case, let me know and I'll go do that, but I want to make sure I've done my diligence.

It started less than a week ago, when my Java app started throwing exceptions and hanging.  I am seeing messages like (sensitive / custom info redacted):

2016-02-01 16:09:55,742 [http-nio-0.0.0.0-9200-exec-22] [<userid>:<username>:REGULAR] ERROR Error executing query against Salesforce: SELECT Owner.Type, Owner.Id, Owner.Name FROM Case WHERE Id='5005000232hgDGNas12'
        at com.myorg.custom.stuff
...
 
Caused by: java.net.SocketException: Connection reset
        at org.apache.axis.AxisFault.makeFault(AxisFault.java:101)
        at org.apache.axis.transport.http.HTTPSender.invoke(HTTPSender.java:154)
        at org.apache.axis.strategies.InvocationStrategy.visit(InvocationStrategy.java:32)
        at org.apache.axis.SimpleChain.doVisiting(SimpleChain.java:118)
        at org.apache.axis.SimpleChain.invoke(SimpleChain.java:83)
        at org.apache.axis.client.AxisClient.invoke(AxisClient.java:165)
        at org.apache.axis.client.Call.invokeEngine(Call.java:2784)
        at org.apache.axis.client.Call.invoke(Call.java:2767)
        at org.apache.axis.client.Call.invoke(Call.java:2443)
        at org.apache.axis.client.Call.invoke(Call.java:2366)
        at org.apache.axis.client.Call.invoke(Call.java:1812)
        at com.sforce.soap.enterprise.SoapBindingStub.query(SoapBindingStub.java:6626)
        ... 302 more
Caused by: java.net.SocketException: Connection reset
        at java.net.SocketInputStream.read(SocketInputStream.java:189)
        at java.net.SocketInputStream.read(SocketInputStream.java:121)
        at sun.security.ssl.InputRecord.readFully(InputRecord.java:465)
        at sun.security.ssl.InputRecord.read(InputRecord.java:503)
        at sun.security.ssl.SSLSocketImpl.readRecord(SSLSocketImpl.java:961)
        at sun.security.ssl.SSLSocketImpl.readDataRecord(SSLSocketImpl.java:918)
        at sun.security.ssl.AppInputStream.read(AppInputStream.java:105)
        at java.io.BufferedInputStream.fill(BufferedInputStream.java:246)
        at java.io.BufferedInputStream.read(BufferedInputStream.java:265)
        at org.apache.axis.transport.http.HTTPSender.readHeadersFromSocket(HTTPSender.java:583)
        at org.apache.axis.transport.http.HTTPSender.invoke(HTTPSender.java:143)
        ... 313 more

I managed to catch the app locked up, and took a thread dump.  I saw long running threads like:

"http-nio-0.0.0.0-9200-exec-3124" #930760 daemon prio=5 os_prio=0 tid=0x00007fbe2406e000 nid=0x2be8 runnable [0x00007fbdb874e000]
   java.lang.Thread.State: RUNNABLE
        at java.net.SocketInputStream.socketRead0(Native Method)
        at java.net.SocketInputStream.read(SocketInputStream.java:150)
        at java.net.SocketInputStream.read(SocketInputStream.java:121)
        at sun.security.ssl.InputRecord.readFully(InputRecord.java:465)
        at sun.security.ssl.InputRecord.read(InputRecord.java:503)
        at sun.security.ssl.SSLSocketImpl.readRecord(SSLSocketImpl.java:961)
        - locked <0x00000006b52dd3d8> (a java.lang.Object)
        at sun.security.ssl.SSLSocketImpl.performInitialHandshake(SSLSocketImpl.java:1363)
        - locked <0x00000006b52dd4c8> (a java.lang.Object)
        at sun.security.ssl.SSLSocketImpl.startHandshake(SSLSocketImpl.java:1391)
        at sun.security.ssl.SSLSocketImpl.startHandshake(SSLSocketImpl.java:1375)
        at org.apache.axis.components.net.JSSESocketFactory.create(JSSESocketFactory.java:186)
        at org.apache.axis.transport.http.HTTPSender.getSocket(HTTPSender.java:191)
        at org.apache.axis.transport.http.HTTPSender.writeToSocket(HTTPSender.java:404)
        at org.apache.axis.transport.http.HTTPSender.invoke(HTTPSender.java:138)
        at org.apache.axis.strategies.InvocationStrategy.visit(InvocationStrategy.java:32)
        at org.apache.axis.SimpleChain.doVisiting(SimpleChain.java:118)
        at org.apache.axis.SimpleChain.invoke(SimpleChain.java:83)
        at org.apache.axis.client.AxisClient.invoke(AxisClient.java:165)
        at org.apache.axis.client.Call.invokeEngine(Call.java:2784)
        at org.apache.axis.client.Call.invoke(Call.java:2767)
        at org.apache.axis.client.Call.invoke(Call.java:2443)
        at org.apache.axis.client.Call.invoke(Call.java:2366)
        at org.apache.axis.client.Call.invoke(Call.java:1812)
        at com.sforce.soap.enterprise.SoapBindingStub.query(SoapBindingStub.java:6626)
...

I am using the Enterprise SOAP API, and nothing in my app code changed since when this started happening.  In fact, while some queries fail, most are succeeding.  Some questions I have:

1. Is there a way to get from Salesforce application logging tracking the incoming API connections / requests?
2. Under what circumstances might Salesforce reset connections?
3. Can I configure connection pool size limits and / or timeouts from Salesforce?
4. Is there any way to track SOQL query performance?
5. Similarly, does my Salesforce instance have any performance monitoring?

Thanks!  Any help is appreciated.