You can certainly solder wires directly under the Arduino, to the pins of the Arduino connectors, and if your soldering does not result in any short there should be no problems after doing so.
However, to understand your stability requirement better: How do you currently connect to the Arduino, such that you are facing stability issues?
Using a breakaway single-row pin header strip, with your wires soldered to the pins should provide a robust enough connection, if you do not want to use a shield or some such. If you use a strip with enough pins to cover the entire row of sockets at each edge of the Arduino, that ought to give you solid mechanical stability. You just solder your wires to the specific pins you need to use, leaving the rest of the pins in the strip unused.
I use pin headers like this one for similar purposes, that way my wires are angled away from the Arduino board and do not obscure any part of it.