Does the paid version of Genymotion support root access and allow writing custom certificates into the system directory?

Yes. Paid Genymotion Desktop supports dynamic root access on Android 12+ and lets you install user certificates that can be turned into system certificates on rooted Android 14+ devices with the Cert-Fixer method. Root can be toggled from the virtual device settings, and the Burp Suite workflow explains that on Android 14+ you must root the device first, then install the certificate and the plugin so it is copied into the system certificate store.

Relevant sources: Genymotion Desktop Release Notes, How to connect Burp Suite to Genymotion, Virtual Devices Settings