How/where do you add fields in uniform?

Should be:

$form = new Form([
        'email' => [
            'rules' => ['required', 'email'],
            'message' => 'Please enter a valid email address',
        ],
        'name' => [],
        'objects' => [],
        'message' => [
            'rules' => ['required'],
            'message' => 'Please enter a message',
        ],
    ]);

For validation, I’m afraid you have to implement your own validator, as there is none in Kirby that accepts an array as value.