Welcome to the forum!
It sounds like you might have a ground loop between your sub and your receiver. I think that might explain the hum that you get when you only use a single connection point. I'm just speculating.
I don't think that connecting to the input jack on the TAPE would hurt anything. I hope others will chime in if I'm wrong.
Also, it won't use any more power from your Sherwood.
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I also want to make sure that we're on the same page for the connection, so here's a bit more.
Based on the title of your thread, I'm assuming that you want to use only the sub portion of your Gigaworks G500 system with your Sherwood (and not the other speakers). If that's not the case, let us know, because the answer will be different than the one that I give below.
To connect the "PRE OUT SUBWOOFER" output from the Sherwood to the G500 sub, you would use the cable that you mentioned. You would connect the 3.5mm jack to the "CENTER/SUBWOOFER" input on the G500 (orange colored, I believe). With that input on the G500, one of the RCA plugs would be for the center channel, while the other is for the subwoofer. If that's the connection that you've made already, then you know that the white RCA connector is for the subwoofer. The red would then be for the center channel, which you don't need to use if you're just trying to send a subwoofer signal. If connecting the other gets rid of the hum, though, then do what you've done (unless someone chimes in and believes it will hurt something).