The US government has essentially zero legal obligation to admit non-residents into the country. The law can and does discriminate against travelers and prospective immigrants based on country of origin, for example. Entry into the United States for foreigners is a privilege, not a right.

That said, a blanket ban on Muslims entering is a brain-dead idea. But I see no good reason why the United States shouldn't consider all information when deciding how to allocate its limited resources for additional screening.

As to your first point, the US government is bound by the constition, and legal opinion is split as to the constitutionality of a ban on all Muslims. http://www.wsj.com/articles/BL-LB-52706

As to the second point, whether or not someone is a Muslim gives you essentially zero information as to whether or not they are a terrorist. (Trivially, it makes them more likely to be an Islamic terrorist, just as Hindus are more likely than non-Hindus to be Hindu terrorists. But since such a miniscule fraction of Muslims are terrorists, applying additional screening to Muslims would make no difference.)