Well another example of course is of Bharat from the Ramayan who ruled as a regent for Shri Raam after installing his wooden sandals on the throne. As mentioned in the following verses of the Ayodhya Kand:
भरतः शिरसा कृत्वा सन्न्यासम् पादुके ततः |
अब्रवीद्दुःखसम्तप्तः सर्वं प्रकृतिमण्डलम् || २-११५-१५
चत्रम् धारयत क्षिप्रमार्यपादाविमौ मतौ |
अभ्याम् राज्ये स्थितो धर्मः पादुकाभ्याम् गुरोर्मम || २-११५-१६
भ्रात्रा हि मयि सम्न्यासो निक्षिप्तः सौहृदादयम् |
तमिमम् पालयिष्यामि राघवागमनम् प्रति २-११५-१७
क्षिप्रम् सम्योजयित्वा तु राघवस्य पुनः स्वयम् |
चरणौ तौ तु रामस्य द्रक्ष्यामि सहपादुकौ || २-११५-१८
ततो निक्षिप्तभारोऽहम् राघवेण समागतः |
निवेद्य गुरवे राज्यम् भजिष्ये गुरुवृत्तिताम् २-११५-१९
Having dedicated the sacred trust to the wooden sandals, Bharata sore-stricken with anguish, spoke to the entire body of his ministers as follows:"Hold the state canopy speedily over these wooden sandals, which have been accepted as symbols of the feet of Rama, my noble brother. By the presence of these wooden sandals of my brother here, a piety has been established in the kingdom. As a mark of compassion, this trust has been placed at my disposal, by my brother. There trust as such will be preserved till his return. Having restored these wooden sandals personally to Rama's feet immediately after his return, I shall behold those feet of Rama along with these sandals. Restoring the kingdom to my elder brother, when re-united with him and thereby laying down the burden, I shall then assume the role of a servant of my elder brother."
Possibly Dhritrashtra could be considered a regent in place of Pandu when he left the throne to go to the forest which is hinted at in Section CIX of Sambhav Parva of Mahabharat:
'The Kuru king, having said this unto his wives, gave away to Brahmanas the big jewel in his diadem, his necklace of precious gold, his bracelets, his large ear-rings, his valuable robes and all the ornaments of his wives. Then summoning his attendants, he commended them, saying, 'Return ye to Hastinapura and proclaim unto all that Pandu with his wives hath gone into the woods, foregoing wealth, desire, happiness, and even sexual appetite.' Then those followers and attendants, hearing these and other soft words of the king, set up a loud wail, uttering, 'Oh, we are undone!'
Then with hot tears trickling down their cheeks they left the monarch and returned to Hastinapura with speed carrying that wealth with them (that was to be distributed in charity). Then Dhritarashtra, that first of men, hearing from them everything that had happened in the woods, wept for his brother. He brooded over his affliction continually, little relishing the comfort of beds and seats and dishes.
There are also examples of human kings such as Nahush ruling as regent of Swarg when Indra had to relinquish the throne after killing Vritra. Vishwamitra also became the de facto regent for the Suryavanshi throne when he forced Harishchandra and his family out of Ayodhya while testing his character.
As for the second part of your question, there is no specific title that is taken up by these regents. Some of these regents retained the titles of the ones they were replacing, for example Dhritrashtra, while others like Bharat did not.