This is a quick thing I'm confused about, so here we go:

Hssho's mining probes mine fidsi's, as well as strategic and luxury resources, and the fidsi's go to your primary system (usually you're original). 

Question 1: These additional fidsi's are only active when the mining drone is active, right?

Question 2: Is the boost per population in my home system, or is it just that number per system?

Question 3: https://imgur.com/a/LBkJjDf is that circled area the boost from the mining operation?

A1: Yes

A2: Effectively per population in home system, it is applied on planet from population there.

A3: No, that is from system Improvements or Heroes. Probe output is specified in a tooltip over icon to launch it:

and is shown in planet tooltip:

A2: Effectively per population in home system, it is applied on planet from population there.

So if a probe is giving me 2 industries per turn, and my home system has 13 pop, that means for the duration of the probe I am getting an additional 26 industry right?

That's right. And those are subject for multiplication from bonuses on planet and system output, so you can end with more then that.