Best way to solder connections between header posts?
You might consider wire wrapping.  

In its era, whole computers and complex semi-custom machines were built this way.  The posts normally used were longer, to permit multiple wrapped connections, but you should be able to get a decent single connection wrapped on a typical header pin.

Durability will really come primarily from mounting of the boards and organizing and securing the wire runs.  If it makes you feel better you can solder the wraps, but it's not clear that will actually help once you have securely mounted the boards and protected the assembly from anything that might snag or pull the wires.

A more formal solution would be to makeup custom wiring harnesses using crimped contacts and multi-channel housings matching an entire header.  You could also consider replacing plain headers with those having latching shrouds, though a multi-contact housing will be a lot more resistant to unplugging than the single-contact ones you are using.