I highly doubt there will ever be parity. There will also be some differences where there can be improvements made over GitHub. Right now you only hear about individuals moving over or open source software. This is mainly because they don't need the enterprise features that GitHub has. I am not sure if you self hosted forgejo if you would get those as well. It's meant to be free. This is the biggest thing that would be missing. For individuals, I don't think you'll notice many differences.

For your second question. It doesn't matter. What matters is git. There are 2 main ways to access this. SSH and http. All you really need is git. There are other tools out there like gitlab. Gitlab is pretty decent too, but the workflow is a bit different. You can even self host with gitlab too. You may not see many direct integrations to gitlab either, but people have been working with it for years now. There is also bitbucket. I think forgejo has a better chance at succeeding though if it continues to do what it's doing. Just tailor to the free and open source community. Really the only thing people need to do is create an account and set up some ssh keys. Beyond creating an account, there is no difference in how you clone a repo from GitHub and codeberg. Just copy the URL and do git clone.

TL;DR GitHub and codeberg have different missions. It will be a similar experience and features for individuals or open source projects are close to the same on both platforms. You probably won't notice much of a difference.