Can a power conditioner cause electrical problems?

Perplexing.

I have a simple test unit for outlets--a plug with 3 lights that light up different combinations of red and green, to tell you if there's juice and if everything is working correctly. 

You possibly could have a wire that is not seated all the way or a screw that needs to be tightened. I noticed this after adding some NCF outlets and one outlet would go out but after plugging the unit in and using a wood dowel, found the loose connection.

Wig

I’m with Wig. Research the connections. 

Check the fuse. There should be one in the IEC receptacle or it may be inside the unit.

There is no problem with either of your products.
What happened:
When yuu first plugged in the new item, something in the way the caps are used.. the capacitors inside it immediately tried to charge, fast. The plugged into component must have a protection circuit which was triggered by the inrush of current.
The most usual type is the "Ground Fault" type. The current pull makes the device think it is a ground fault and shuts off.

When you plug the new device in a second time, the caps are already partially charged, and they do not cause the duplex plugged into  to shut down.

This happens with a few items, when they have not been used for a long time, or are brand new.

So IMO nothing is wrong at all.
The first time it happened to me, I was surprised too. But once I figured out the reason.. no problem.

PS I came back to the forum JUST TO ANSWER this question.
I have been gone a long time. but decided I had the answer no one else had.
Good luck