Enable texture repeat under "CanvasItem → Texture → Repeat → Enabled".
See also #28.
Yes, partially. The plugin includes a web build, but since it uses GDExtension, a web export template with GDExtension support is required to use it.
According to the Godot documentation for web:
"The default export templates do not include GDExtension support for performance and compatibility reasons".
That means in order to use the web version of this plugin, you will need a custom web export template compiled with dlink_enabled=yes.
To compile it yourself, refer to the official Godot documentation for further information: Building from source, Compiling for the Web and Exporting for the Web.
Alternatively, you can download prebuilt export templates from another trusted source. For example, the LimboAI addon provides web export templates with GDExtension support.
This is a known issue and related to execution order.
TL;DR: Connect to the NativeRopeServer.on_pre_pre_update signal and run the rope update code there. It is essentially an alternative to _physics_process() that runs just before ropes get updated.
See Execution Order Documentation for a more detailed explanation.