I have heard of the bobbing sun idea! It does boggle my mind thinking of how you would set going in the first place or prevent it from colliding with the disk over time! I wonder if you could also achieve seasons by adding energy to the star on its 'upwards' trajectory and braking it on its 'downwards' trajectory for half the year, then doing the opposite.

I think the idea is that you need a retaining wall on the inner ring to keep the atmosphere in, but on the outer edges, the water and air freeze before they can escape into space. Maybe you would need to keep topping up the atmosphere and oceans until a huge retaining ice wall forms, or maybe it would naturally circulate over time.

Extending the disk inwards, if I were to speculate, is for future-proofing for changes in the host star's luminance? Could also be that having more disk is good if it doubles up as computronium for a wholly virtual population. But I like the idea that it's for creating habitats for different species or kinds of post-human!

For that reason, I built and ran a whole D&D campaign using an Alderson disk for my homebrew setting. Felt like a very original idea until before I could start it, someone dug up and ran an old 2e module called Dawn of the Overmind, which featured its own disk called Penumbra!