1- The largest source of cockpit conflict is miscommunication leading to misunderstanding. Any breakdown in communication can lead to conflict up to and including the type that leads to an accident. Ensuring that both crewmembers are well informed of each others actions and though processes is the first and largest step (IMHO) of effective CRM.

2- What does the emergency/abnormal checklist say? All of the procedures that I am familiar with say something along the lines of: Check pressurization. Monitor and continue flight as appropriate. Odds are that the issue is a simple proximity system issue, and not something worth diverting over.

3- I enter icing conditions all the time in flight and don't really care about them. Unless the aircraft isn't certified for known icing or isn't properly equipped (IE deice/anti-ice inop), no biggie.

4- This is probably the most severe of the 4 questions you listed simply because the potential for negative media is possible. I make the assumption that you're sober (you DO follow bottle to throttle rules right?) Best answer? Follow whatever established procedure the company has regarding accusations of intoxicated crewmembers. Start by calling your chief pilot and explaining the situation.