Should a trace be connected through multiple pins on a pin header?
You don't need traces for mechanical strength. The copper inside and around the hole make up all of the mechanical strength. Adding a trace will not make any difference. 



I don't know what you are going to 'hang off' your connector but even a 2x5 header is amazingly strong. Just try to pry one off board: you will need a set of strong pliers. Going to e.g. 20 or 25 pins does not add much. 
For mechanical stability it would be better to use two, e.g. a small one on each side of the board.

Signal integrity depends very much on frequency. If you have high frequency signals you should use connectors with the right impedance. In that case you should not route those signals through multiple pins. Also route direct from the source to the connector. Avoid T-junctions. 

Where multiple pins help, is when you want to void cross-talk. Keep them apart. If your connector goes to a flat-cable you might want to use signals on all even pins and ground on all odd pins. On your cable you then get signal-ground-signal-ground.