How could this GPIO-controlled LED status blinker be improved?
Aside from the fact, that you could save parts but not money by using something like a ne555, it is an interesting circuit. But as you want some hints:
your third transitor will be in saturation, when activated. Depending on the type it may fail to lower the base voltage enough to prevent the right 2n3904 from conducting. If you want to be sure you should use an low VCEsat BJT. NXP and ONSEMI each have a special selection of those types. Then you may increase the resistors of your added voltage divider, but only to save some fractions of miliamperes. 

So why does your circuit react to touching in this way? I bet your powersupply is causing this phenomenon. Either your RPi or the laptop connected to via usb or network is powered by a wallwart or brick PSU. Those devices tend to have a Y-capacitor between hot and cold on primary side. the middle pin is connected to ground on secondary side. This leads to the fact, that the ground of your RPi in fact has a voltage of half of your mains voltage relatively to earth. So if your mains voltage is 110V AC, your RPi's ground is at 55V AC. When you touch it, there will flow a small AC current through your body and the capacitance it forms with your environment. this current is now rectified by the base-emitter-diode in your transistor. It apparently is big enough to open up that transistor and stop the astable circuit from being astable.