As to the first sentence, I think "that" is often needed.
The Google results:
A.” It is important he” 19600
B.”It is important that he" 86500
While the 19600 results include large number of "it is important, he". And the verbs in the rest are not in Subjunctive forms with very few exceptions.
As to the second sentence, I think "that" is optional. And a