Admin/owner permissions - you don't need extra or special permissions to access any Typeform resources. However, for forms created in Shared Workspaces, the connected account needs to be the owner of the Shared Workspace to access the form in the dropdown menu of the Zap Editor (or use the workaround outlined below).
Paid/versioned account - any plan (free or paid) and any version of Typeform can access the integration.
Custom field support - Yes, custom fields (question types) are supported.
Partner's documentation: More information can be found here.
Setting up the Typeform Integration
When you start creating a Typeform Zap, you will be asked to connect your Typeform account.
You’ll now be asked to log into your Typeform account (unless you are already logged in).
Next you will be asked to let Zapier have access to your Typeform account by clicking "Accept".
If all steps were successful your Typeform account will now be successfully connected.