I'll address your last point first. If you don't have a TV involved, then yes, only component cables connected from your cable box to the Slingbox component IN are needed, and you should have video on your Slingplayer apps. When we get a TV involved, difficulties may arise in the form of HDCP (High Bandwidth Digital Copy Protection) when you have an HDMI cable connected from your cable box to the TV.
There are two ways around this should you come across the issue:
1) Disconnect the HDMI cable entirely, and use a second set of component cables from the Slingbox out to the TV.
2) If you wish (or need) to continue using HDMI with your TV, you can use an HDMI-component converter. Here is the one we recommend (though you may still be able to find others that work):
The way this one works is that you connect HDMI from the cable box to the converter, and from the converter, you go both HDMI OUT to the TV, and component OUT into the Slingbox component IN. Now, I'm not sure if your HDMI > Component converter worked the same way, so I'm not sure what could've gone wrong with yours, but we know that the one we recommend works to get video on the Slingbox and TV at the same time.
Just for reference, this article discusses copy protection in relation to Slingboxes (don't mind the title of the article):
Hope this helps!