The answer is yes.
Got a super quick answer here:
I couldn't get it to work just configuring in package.json but this did work:
"test:unit": "sfdx-lwc-jest -- --reporters='jest-tap-reporter'"
On reflection I'm going to use ...
Most records in an org are not visible to tests. You need to insert your own records in your test.
Why? This makes tests self-contained rather than them relying on pre-existing data. It also usually makes assertions easier to write because the data values added are close at hand in the test.
See e.g. Create Test Data for Apex Tests.
The global modifier doesn't change the fact that your execute() method is still an instance method.
ClassName.method() is how static methods are called.
Since execute() doesn't have the static modifier (and I don't think it should have that modifier anyway), you'll need to create an instance of your class to be able to call the instance method.