Can I use multiple external drives with Apple Studio Display?

First: Studio display's only Thunderbolt port -- will output enough power to M3 MBA's Thunderbolt 4 port. Therefore, you do not need to use power adapter for the MacBook Air while connect to the Studio Display.

That will leave you: 3 USB-c ports on the Studio Display and ONE Thunderbolt 4 on MacBook Air.

The USB devices will be OK via adapter (USB-A to USB-C).

However, you mention the 1x2T Thunderbolt drive with older "Thunderbolt -> USB-3 converter" - which can be iffy, regarding the power consumption. The Studio Display can be limited for the voltage when too many devices were connected to it (my personal experience).

Option 1: You can get Thunderbolt 3-> Thunderbolt 2 adapter, and connect to the MacBook Air directly (I am not sure, since MacBook Air uses power from Studio Display.

Option 2(preferable) : Get a decent Thunderbolt Dock like this, which can be very diversified for many peripherals.

Back to your question "Or will I need to replace the drives with SSD versions with USB 3 connectors?"

That might be a good idea, that newer external drives are equipped with more up-to-date connector (USB-C).

Best

Best way to find out is to purchase one and test within the 14 day no-questions-asked return window. It’s meant exactly for the type of questions you have, so use it. Good luck!