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.
Roku 3, 42" LCD, Ethernet (6.2)
Roku Stick HDMI, 32" LCD, WiFi (6.2)
Roku2 XS, 25" CRT, Ethernet (6.2)
Router - ASUS AC68U
ISP - TWC - 50 Mbps/5 Mbps