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You can use a PS4 controller with PS5, but you can only use your PS4 pad to play PS4 games on PS5.
Makes sense since it would be like trying to play Wii Sports with a GC controller, although some ps5 games probably work fine without dualsense
You can use a ps5 controller on ps4 via ps4 remote play (just to get the inputs to the ps4, don't need to actually play on pc). It's definitely not as friendly, but it's not all impossible.
There's nothing from stopping you picking an external PS5 SSD though. You can store and play PS4 games from here (so much faster than an HDD/regular external hard drive). And as of an update on April 15 2021, you'll finally be able to store PS5 titles externally too. 

BTW I wouldn't call an expensive proprietary expansion card consumer friendly. But at least it was available quickly.
They're CAD 300 here, 80 dollars less than a Series S. I would call that an anti-consumer move.