The amp is likely wired in parallel, so either two, 8 ohm cabs in one output, or one 8 ohm cab in each output will still be 4 ohms and result in the maximum power. If you have two 8 ohm cabs in one output, you cannot add another cab to the amp, as the impedance will be too low (< 4 ohms). "IF" the amp was wired so that the cabs were connected in series, (not likely at all), they would create a 16 ohm load and a very low power output.
So yes, if you have one 8 ohm cab in each plug that results in the 4 ohm minimum load that the amp can handle. If you daisy chain the cabs into a single output (amp-> cab 1-> cab 2) they will also create a 4 ohm load. Once you have reached the 4 ohm load, don't add any more cabs to the amp!
How are you connecting the Sunn combo to the Hartke? Does the Sunn combo have an input that allows you to connect directly to the internal speaker, bypassing the amp? If so, then you could feed the Hartke signal from one of the outputs to the Sunn speaker, but the impedance of the Sunn speaker has to be considered when looking at your ohm load on the Hartke. What impedance is the speaker in the Sunn? Some combos are already 4 ohm to get the most out of the amp. If the Sunn has an 8 ohm speaker then you can only connect one more 8 ohm cab to the Hartke. If you double up two 8 ohm speakers, or a single 4 ohm cab on the remaining output, you would be running a 2.6 ohm load, (too low!) and that could damage the amp. If the Sunn does not have an input that goes directly to the speaker, you cannot drive the speaker that way.
An alternative would be to use the Sunn as a slave amp, meaning you have the Sunn amp turned on and are using the signal from the Hartke as the input signal to the Sunn. To do this you have to take the signal from the preamp section of the Hartke. There appear to be only two options for accessing the preamp signal from the Hartke, either from the DI out, (which would be problematic because that output is an XLR connector, and also is a quite low level signal -10dB, optimised for a sound board input), or from the Effects Send on the back of the amp, which can feed an external amplifier that can accept a +4dB signal (this would appear to be the most likely route). Either of those signals would connect to the input of the Sunn. You would then use the Sunn amp volume to balance the volume between the two units.
You can not take the powered, speaker out signal from the Hartke and feed it to the amp input in the Sunn. That signal would be way too powerful for the Sunn input.
I have no idea why any lights should come on in the Sunn if it isn't plugged in and you are simply feeding the Hartke signal direct to the speaker itself. I have no idea what might happen if you fed the Sunn amp input with Hartke powere speaker output.
If the Sunn does not have an input to its speaker that bypasses the amp, then you can't feed that speaker with a speaker out from the Hartke. You'd have to try to use the Sunn as a slave amp.
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