Is it permissible for the Imam to supplicate aloud for the deceased, the sick, and the traveler in the mosque, with the congregation saying "ameen," and is it legislated to raise one's hands during this supplication?

Requesting prayers from living righteous individuals is permissible by the consensus of the Ummah. However, the Imam should guide people towards what is better, which is for them to strive in supplicating for themselves. He should also clarify the detrimental effects of asking creation, such as dependence on other than Allah, harming the one being asked, and debasing oneself. As for collective supplication for a traveler or someone in need, it was not part of the guidance of the Prophet (peace be upon him) nor his companions. Therefore, it is better to guide the one requesting prayers to supplicate for himself. There is no harm in individually asking for prayers for him in his absence. If the matter is an urgent, unforeseen circumstance, there is no harm in requesting prayers from a gathering, but habitually doing so is closer to innovation (bid'ah).