I sadly can't confirm any of these problems. I have done daily quests (which include "get rank SS" and "clear chaos f3" and so on in mission maps) in a party without any problems. Everyone of us was able to solve their particular quests, for example I solved "chaos SS" while my party member solved "chaos normal" - we did two maps chaos for it - without any problems.
Btw if your party member does not have the quests you have, or does not have matching quests only the maps needed to solve the quests for the party leader are coming up in a mission map. For example if the party leader has the quest to kill sheep in the windia plains mission map, while all others have to kill wolves inside of the windia plains mission map only the map with the sheeps inside will come up. Because the map coming up is chosen based on the currently running quests of the party leader. This was always the case and is completely normal. If you are on another quest inside the same mission map the party leader just has to pass on the party leadership to you to make you able to complete your quest too.
You are not manually ported out of mission maps in the case the party leader leaves, you have to press the leave button manually. I believe this is the case since at least the Galaxy update came into the game.
Just for your information, there is no penalty. There used to be in the past, but the only thing still in the game is the text. Maybe it is about time to remove the text too.