The Soyuz family has many variants. Do any of them require different launchpad infrastructure? If multiple active variants require different infrastructure, how are the logistics of that handled? Do they use different pads?
Yes, there are two major differences between the traditional launch pads at Baikonur and Plesetsk and the modern launch pads at Kourou and Vostochny:
The analog attitude control system of older R-7 / Soyuz launchers could only change the pitch angle during flight - yaw and roll were fixed (but controlled). This means that to reach different inclination orbits, the launch pad has to be rotated into the correct roll angle for that inclination. Soyuz 2.1 has a digital attitude control system, which can change all three angles, so that launch pad rotation is not necessary with pads which are only designed for these newer models (Kourou, Vostochny).
At Baikonur and Plesetsk, the vehicle is completely integrated horizontally in the vehicle assembly building and the complete vehicle is then rolled out to the launch pad. For Soyuz ST at Kourou, ESA requested vertical integration of the payload. The rocket is rolled out to the pad without payload. After the rocket is erected, the payload is integrated in a mobile service tower. The tower is rolled back to a safe distance before launch. Vostochny uses the same design.