Roku 3 does not have RCA outputs and the Digital Converter box will not help with your situation (actually totally ignore it when it comes to Roku, neither have anything to do with on another), as you mentioned it has one HDMI "OUT", you would need an HDMI "IN". Recommend a Roku 1 or 2 instead for your situation.
Roku2 XS, 42" LCD, Ethernet (5.5 b320)
Roku Stick HDMI, 32" LCD, WiFi (5.5 b2027)
Roku N1000, 25" CRT, Ethernet (3.1)
Router - Linksys WRT54G2
ISP - TWC - 50 Mbps/5 Mbps