[SalesForce] Connection Reset Exception using SOAP Integration

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] [152031:username:REGULAR] ERROR Error executing query against Salesforce: SELECT Owner.Type, Owner.Id, Owner.Name FROM Case WHERE Id='5005000000hgDGNAA2'
        at com.myorg.custom.code.SalesforceUtilImpl.executeQuery(SalesforceUtilImpl.java:123)
        at    ...

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, this problem is very intermittent; 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 that might give hints on being overworked?

Sorry if this was asked already. I couldn't find an answer among the other questions that were similar to this one.