The items in the forum that you obtain by searching for $tls_allowed all refer to attempts to stabilize connections with SMTP servers but they involve versions 2.11 and especially 2.13 so they don't apply to your case.
In answer to your question about whether WeBWorK actually checks whether an email has been successfully delivered the answer is no. In recent versions there is some extra checking to make sure that the connection to the SMTP server was successful but nothing to check that the mail was actually delivered.