• Will the the computer's video card transmit the sound over the HDMI cable too? It has a cheap AMD video card; I think it's a Sapphire model HD5450.
It should, but that doesn't mean it will. Some video cards require fiddling with the sound output destination settings a bit. Others will switch automatically.
• Is there some other way to hook these up using two HDMI ports plus component video? I need to look at that Sam's TV and see if it has audio inputs.
Make sure the TV has discrete component/HDMI inputs. Some sets will treat one input as either/or, not both. In other words they may have both HDMI and component on one input designation but you have to choose one or the other. They can't be used at the same time.
• I assume a separate Roku device is better than getting one integrated into the TV? (supports more apps and can be updated) Among other things, I want to run the Spectrum app that I think will let me watch everything I'm currently subscribed to on cable -- until I'm ready to cut the cord which is the ultimate goal but at least a year out (I need to prove the concept to the Mrs. -- I think Sling + Amazon + Netflix + over-the-air channels gives us everything we watch)
From what I've experienced, apps native to a TV/disc player are, at best, little better than "just okay". Slow, difficult to navigate, and often with several glaring deficiencies due to compromises made for the app to integrate into the TV's operating system. For media streaming a Roku box or gaming console are much better. The native Roku interface is simple and intuitive. The only downside is it can be a little slow because of the move-and-select typing system for custom text input. I have an Xbox with the chatpad controller add-on. Short of a keyboard and mouse, it's hands-down the best input combination I've used on any type of streaming device. (FYI - With the introduction of the Xbox One X, One S's are going for $199, new.)
Also, if you've considered a home theater setup, keep in mind that most HT receivers are also functional A/V switches. Most have at least four HDMI inputs plus one or two component inputs that will upscale to 1080p (the more recent units, 4K).
Brad