Linux – Display IP & TCP statistics per interface (netstat -s with interface)

I prefer lsof because it's output is consistent across all platforms on which it runs. You can pretty much get the same info from both programs, though. I think it comes down to personal preference.

1 outgoing packets dropped

There is nearly no packet loss, which is good, but we don't have latency data. At a glance, i'd say you're using the wrong tools for the job.

Is there a database involved? Are there some kind of cyclical functions that slow the system down around the 10 minutes mark? Does the machine only operate this tcp server or is serving other resources?

Netstat is not a proper metric for what you want to do. To be sure your web-application is performing as intended, you need an infrastructure in place featuring the following

• Hooks into your application to ensure proper metrics. You're the developer, so you can do this and it will ease your work massively. By hooks i mean facilities to fetch diagnostic and performance data, coded directly into your application.
• A graphing/monitoring infrastructure. Cacti and Nagios are an example i'm familiar with, but there are more.
• A plan. What do you want to achieve? What level of service do you want to supply your users with? Implement diagnostics and performance metrics as you develop your application and if you get wind this could turn into something big, make it scalable. *Really* scalable.