Soldering small connector to a wire
Just go ahead and wire it to the small connector. The pin spacing is wide enough you shouldn't have any issues with bridging or shorts. Pre-tin the wire and the connector, and add on each one. You could strip each wire a bit longer and twist it around the connector pin if you feel it is not strong enough.

A small piece of heat shrink is all you need to cover up the small connector afterwards. If you are really worried about shorts, you can add heatshrink to each individual wire before soldering it on, then still add the bigger heatshrink afterwards.

An alternative way to attach it would be by using a small piece of protoboard. Solder the connector to the board, then use wires from the protoboard to the main board.