At the end of Kaliyug, the 10th and last incarnation of Vishnu is said to come as Kalki. Are there incarnations after Kalki, at the end of Dwapara yuga after this age? Do the incarnations "start over", or could they be expected to be completely new ones unlike the ones before?
We know at least 7 incarnations of Vishnu after Kalki. Let me explain. Satya Yuga, Treta Yuga, Dwapara Yuga, and Kali Yuga together form one Mahayuga. And 1000 Mahayugas form a Kalpa. A Kalpa is also divided into 14 Manvantaras, each consisting of 71 Mahayugas. We are living in the 28th Mahayuga of the Vaivasvata Manvantara, which is the 7th Manvantara of the present Kalpa. Now as I discuss in this question, we don't have much information about the future Mahayugas of the Vaivasvata Manvantara. But we do have information about future Manvantaras: this chapter of the Srimad Bhagavatam lists seven incarnations of Vishnu who will occur in the remaining seven Manvantaras of the present Kalpa:
O King, during the eighth manvantara, the great personalities Gālava, Dīptimān, Paraśurāma, Aśvatthāmā, Kṛpācārya, Ṛṣyaśṛṅga and our father, Vyāsadeva, the incarnation of Nārāyaṇa, will be the seven sages. For the present, they are all residing in their respective āśramas. In the eighth manvantara, the greatly powerful Personality of Godhead Sārvabhauma will take birth. His father will be Devaguhya, and His mother will be Sarasvatī. He will take the kingdom away from Purandara [Lord Indra] and give it to Bali Mahārāja.
O King, the ninth Manu will be Dakṣa-sāvarṇi, who is born of Varuṇa. Among his sons will be Bhūtaketu, and Dīptaketu. In this ninth manvantara, the Pāras and Marīcigarbhas will be among the demigods. The king of heaven, Indra, will be named Adbhuta, and Dyutimān will be among the seven sages. Ṛṣabhadeva, a partial incarnation of the Supreme Personality of Godhead, will take birth from his father, Āyuṣmān, and his mother, Ambudhārā. He will enable the Indra named Adbhuta to enjoy the opulence of the three worlds.
The son of Upaśloka known as Brahma-sāvarṇi will be the tenth Manu. Bhūriṣeṇa will be among his sons, and the brāhmaṇas headed by Haviṣmān will be the seven sages. Haviṣmān, Sukṛta, Satya, Jaya, Mūrti and others will be the seven sages, the Suvāsanas and Viruddhas will be among the demigods, and Śambhu will be their king, Indra. In the home of Viśvasraṣṭā, a plenary portion of the Supreme Personality of Godhead will appear from the womb of Viṣūcī as the incarnation known as Viṣvaksena. He will make friends with Śambhu.
In the eleventh manvantara, the Manu will be Dharma-sāvarṇi, who will be extremely learned in spiritual knowledge. From him there will come ten sons, headed by Satyadharma. The Vihaṅgamas, Kāmagamas, Nirvāṇarucis and others will be the demigods. The king of the demigods, Indra, will be Vaidhṛta, and the seven sages will be headed by Aruṇa. The son of Āryaka known as Dharmasetu, a partial incarnation of the Supreme Personality of Godhead, will appear from the womb of Vaidhṛtā, the wife of Āryaka, and will rule the three worlds.
O King, the twelfth Manu will be named Rudra-sāvarṇi. Devavān, Upadeva and Devaśreṣṭha will be among his sons. In this manvantara, the name of Indra will be Ṛtadhāmā, and the demigods will be headed by the Haritas. Among the sages will be Tapomūrti, Tapasvī and Āgnīdhraka. From the mother named Sunṛtā and the father named Satyasahā will come Svadhāmā, a partial incarnation of the Supreme Personality of Godhead. He will rule that manvantara.
The thirteenth Manu will be named Deva-sāvarṇi, and he will be very advanced in spiritual knowledge. Among his sons will be Citrasena and Vicitra. In the thirteenth manvantara, the Sukarmās and Sutrāmas will be among the demigods, Divaspati will be the king of heaven, and Nirmoka and Tattvadarśa will be among the seven sages. The son of Devahotra known as Yogeśvara will appear as a partial incarnation of the Supreme Personality of Godhead. His mother’s name will be Bṛhatī. He will perform activities for the welfare of Divaspati.
The name of the fourteenth Manu will be Indra-sāvarṇi. He will have sons like Uru, Gambhīra and Budha. The Pavitras and Cākṣuṣas will be among the demigods, and Śuci will be Indra, the king of heaven. Agni, Bāhu, Śuci, Śuddha, Māgadha and others of great austerity will be the seven sages. O King Parīkṣit, in the fourteenth manvantara the Supreme Personality of Godhead will appear from the womb of Vitānā, and His father’s name will be Satrāyaṇa. This incarnation will be celebrated as Bṛhadbhānu, and He will administer spiritual activities.
So the seven known incarnations of Vishnu after Kalki are Sārvabhauma, Ṛṣabhadeva, Viṣvaksena, Dharmasetu, Svadhāmā, Yogeśvara, and Bṛhadbhānu.
To answer your side question, some incarnations of Vishnu do recur. For instance I discuss in my question here how there have been three Vamana incarnations in the present Kalpa. And I discuss in my question here how Kalki came once before in the present Kalpa.