do I now need to use the H9 software for updating my SPACE pedal (and ditch the previous FactorLib, etc.)?
Well, the actual application that you use now for updating your Space is called the Eventide Direct Updater. You can download and install that from the Space support page, or if you want to use H9 Control with your Space, it comes with the Eventide Direct Updater bundled into it. So, if you connect H9 Control to a Space that doesn't have v5 software installed on it, H9 Control will offer to launch the Eventide Direct Updater for you to update your Space.
Once you install v5, you'll need to start using H9 Control with your Space as FactorLib won't work with it anymore…
Also, is the 5.0.0(5) update simply to enable compatibility with the H9, or are there other tweaks in it?
It was primarily an update to make Space work with H9 Control. There were some tweaks to it but the algorithms that come with Space and the parameters you have to control them didn't change.
And on the specific SPACE update page, I don't see that 5.0.0(5), unless that's what will come up using the H9 software.
When you open up the Eventide Direct Updater with your Space connected, it will show you the list of available software for your Space including the latest beta software and the earlier Space software releases.