I've never tried exactly what you are talking about so I don't have direct experience with that, but I've used landscaping cloth for other things. I assume landscaping cloth is what you are talking about when you say weed barrier. Part of the answer will depend on how deep your raised bed is and what you plan to grow in there, thinking you will be limited to shallow rooted plants. I would think that one cinder block thick would not be deep enough for much of anything, other than maybe radishes. It is conceivable that you would do some good, but I don't think it very likely and I would not do it. The stuff I've used is supposed to allow water through, so I don't think you would need drain holes. But it will hinder root development on your plants if their roots go that deep. I think newspaper would also hinder root development until it broke down. I'd be hesitant to use newspaper too. I once dug up a section of newspaper in a New Orleans suburb that had been buried about a foot deep for seven years and I could still read it. I was truly amazed. In that climate I would have expected that newspaper to break down pretty quickly. I was very wrong.
I think what you are after is to put something on top of the sod so you don't have to dig it out. If you use good quality landscaping cloth with no openings in it, it would probably work, but I don't think the cheap stuff will totally stop a grass like Bermuda or St. Augustine from finding a way through. You probably don't have the time but one way that might be effective is to put heavy black plastic down and let it cook under the hot sun a couple of months to try to kill it out. Usually I dig it up and remove what grass and roots I can, but I never get it all. If you leave one small piece of grass with one of those nodes on it, it will sprout again. If you get the big pieces of grass and the roots out, it is not all that hard to pull these sprouts out if you get them while they are young. But if you leave long pieces of those runners or clumps of roots, you'll have to dig those out or they will keep sending up shoots. I strongly dislike my Bermuda grass around my gareden and such.
I've used it under mulch around trees when I planted them. It works fine for a while, but then my Bermuda grass runs under it and comes up in any opening. Or it runs in under the mulch on top of the cloth and sinks its roots in the cloth. When I pull the grass out, the cloth comes up with it. It helps a while and it makes it easier to pull that grass out, but now I just mulch and don't use the landscaping cloth. I just use a lot of mulch and that makes it reasonably easy to get the grass and weeds out. I also use newspaper under that mulch. The newspaper breaks down but does help keep the Bermuda grass runners from poking through as much.
I've used landscaping cloth in landscaping beds, again under mulch. Same thing with the Bermuda grass. It will run under the cloth for a few feet and come up through any weak areas. It does make it easier to pull that grass out, but I have to take the mulch off, take up the landscaping cloth, and remove those runners and anything that comes through. That stuff does have its uses and it can make life easier, but I don't consider it a permanent solution. It is a tool and a temporary barrier.
Edging also has its uses. It defines your bed and gives you something to weed whack against. As long as you don't have chickens to scratch in it or dogs to dig in it, edging can help contain your soil or mulch. But as a barrier to Bermuda grass, no. Maybe if you had a solid barrier buried a couple of feet thick, but just a few inches stuck in the ground is not much of a hindrance.
I have made life easier by cutting a strip of landscaping cloth about 12" to 16" wide and putting it under the edging with the excess sticking into the bed, running back up to the top of the soil but under the mulch. This does not stop the Bermuda grass but when I remove the mulch and the cloth, it makes it easier to dig out those runners from the Bermuda grass. It probably does not help enough to make it worth the effort, but I sincerely dislike that Bermuda grass and do try different things.
