As for ad hominem, your example it does connect the characteristic to the correctness (albeit falsely). Their example is just making a random insult against anyone who has a certain opinion. Had they said "We shouldn't allow the mosque because those that want to are America-hating liberals" that would have been a rather poor example of an ad hominem fallacy. As it is their construction does not even look like it relates to a debate.
You make the same mistake with appeal to authority, albeit with a better example. Imagine I say that tinfoil is best because babies lose heat through radiation. Saying that a soft blanket is better because Dr Spock says so is a fallacy. A valid argument would be to outline Dr Spock's reasoning.
I just noticed they got begging the question completely and utterly wrong! Begging the question is making an argument that assumes the conclusion or simply restates it. I cannot even twist their example to actually beg the question, so I will nick Wikipedia`s example:
"Opium induces sleep because it has a soporific quality"