How you quilt it is up to you and the kind of batting you use. You can do the blocks, but if your batting needs a line of stitching every 2 inches, you'll have to do some of the logs.
I have never used the Connecting Threads thread. I use hand quilting thread - it's glazed so it doesn't fray as quickly. There's no reason you can't use other thread, but you might want to use short lengths, and run it through Thread Heaven to keep it from fraying.
But most of all, have fun quilting. :)
Janet