Yes, the current Brahma attains Moksha at the end of his life. As I discuss in this answer, the Srimad Bhagavatam describes what happens to the inhabitants of Brahmaloka, including Brahma himself, at the end of the Mahakalpa:
Worshipers of the Hiraṇyagarbha expansion of the Personality of Godhead remain within this material world until the end of two parārdhas, when Lord Brahmā also dies. After experiencing the inhabitable time of the three modes of material nature, known as two parārdhas, Lord Brahmā closes the material universe, which is covered by layers of earth, water, air, fire, ether, mind, ego, etc., and goes back to Godhead. The yogīs who become detached from the material world by practice of breathing exercises and control of the mind reach the planet of Brahmā, which is far, far away. After giving up their bodies, they enter into the body of Lord Brahmā, and therefore when Brahmā is liberated and goes to the Supreme Personality of Godhead, who is the Supreme Brahman, such yogīs can also enter into the kingdom of God.
In my question and answer here, I discuss a couple more quotes, proffered by the ancient sage Badari, in defense of the view that the inhabitants of Brahmaloka attain Moksha. First of all, there is this verse from the Mundaka Upanishad:
Having well ascertained the object of the knowledge of the Vedânta, and having purified their nature by the Yoga of renunciation, all anchorites, enjoying the highest immortality, become free at the time of the great end (death) in the worlds of Brahmâ.
And there's verse 12.269 of the Kurma Purana:
When the pralaya has come and the end of the Highest, they all together with Brahma enter the highest place.
Note that the inhabitants of Brahmaloka don't just automatically attain Moksha as a birthright. Rather, it is through meditation on the supreme Brahman while they're in Brahmaloka that they ultimately attain Moksha. See my question and answer here.
To answer your other question, yes, when a new Mahakalpa begins, someone else is born as Brahma. I'm not sure whether Hanuman being the next Brahma has a basis in Hindu scripture, but a question was asked about it here.