Is it possible to somehow trigger case assignment rules upon creation of a case via a lightning-record-edit-form? It used to work when I had it set in page layouts but that seems to no longer be the case.
There's no direct way of running the assignment rule if you are using
lightning-record-edit-form. With little customization, you can though achieve this.
You can introduce a custom checkbox in your component similar to what is on standard page layouts. Then in after trigger of Case, you can fetch that record, verify if the checkbox is selected, run the assignment rules as described on Run Case Assignment Rule from Apex.