There always needs to be a nominated captain on the field for the bowling team as certain communications are specified in the rules to occur between the Umpires and the captain. If the captain is off the field due to injury or some other reason, generally the vice-captain takes over the official role while the captain is absent, but it doesn't have to be him, any other player would be permitted to act as captain. This often occurred when Hashim Amla was vice-captain of South Africa. Even when the captain was absent, someone else took over and he remained vice-captain.
There is nothing official in the rules about the role of the vice-captain, and what they actually do in practice varies between different teams. Some teams don't have a vice-captain at all and instead have just a 'leadership group' of senior players who help the captain by suggesting tactics, field placements, bowling changes etc.
As for team selection, whether or not the captain themselves has a say varies considerably between different teams, depending on how they structure the selection committee. Whether the vice-captain has any role would also vary, though in most cases they would not play any role in selections.