How can I control the connection/disconnection of 8 bi-directional logic level lines, one 5V-50mA power, and a ground line with one SPST switch?
You also can use 10 transistors, you connect the SPST switch to the base of each transistor. Than each collector/emitter pair takes care of the current to be flow for each channel. Most transistors will be able to manage 50 mA easily.

There are ICs too for this, but probably not with 10 lines but (max) 8, so you need 2 ICs (transistors might be easier in this case). 

Relays are overkill for just 50 mA each.

If the devices are all using the same voltage (or most of them), you can also run them in parallel possibly.

Read https://nl.wikipedia.org/wiki/Transistor about resistors.