TestCafe - How to check if a web element exists or does not exist without failing the test?

You can check the async exists property in the if condition in the following way:

if(await things.thingSelector(thingName).exists) {
    // do something 
} 

You can use the following assertions to test existence and non-existence elements:

test('Test existence and non-existence elements', async t => {
    await t
        .expect(Selector('#existing-element').exists)
        .expect(Selector('#non-existing-element').exists).notOk()
});