Can I set pin 13 to output 5v, then put arduino mini pro to sleep? Can I set pin 13 to output 5v, then put arduino pro mini to sleep and it maintains 5v going through pin 13 while in sleep mode?

I know there are various "levels" of sleep - can a voltage be maintained on a pin for all/any of the sleep modes?
Given the absence of clarity on the driving state of pins during sleep, consider connecting Pin 13 to an SR Latch or the gate of a MOSFET.

An SR Latch will hold it's output high or low until a high is presented on the Set or Reset inputs, so connect Pin 13 to the Set input, with a pull-down resistor -- even if the MCU in sleep mode allows the voltage output to drain away to 0v, the SR Latch will maintain it's output.

A MOSFET gate has a non-trivial capacitance which can hold a charge if the gate's driver goes high-impedance, thus holding the gate open after the driver is disconnected.