I have been repairing iphone's and ipod's for several years. Most of my screens and parts have been purchased from ebay, and I have never had any issues with them, nor has my customers. To be fair they are not the same quality as the official screen's but what 3rd party parts are.
The home button may not be working as part of the screen fault, you said it was quite bad. The only way to tell is to replace the screen first.