On March 10, 2016, Jeremy Crawford tweeted, "No rule lets you opt to fail a save. As DM, I might allow it, assuming you aren't incapacitated or dominated." Mike Mearls also said that he would allow it (as a DM).

Willingly failing a saving throw is a house rule that the designers are okay with, but it is a house rule. Strict RAW does not allow it.

Yes

The effect is suppressed, but it is not ended. Frightful Presence specifically says that you can repeat the saving throw at the end of each of your turns. Nothing in Calm Emotions changes that. If the spell had another part, you wouldn't say that you don't get the saving throw. I don't think it makes any sense to say that you don't get the saving throw since the only effect is suppressed.

Let's change the wording slightly:

Cold: Each creature of the plaguebearer's choice that is within 120 feet of the plaguebearer must succeed on a DC 16 Constitution saving throw or become sickened for 1 minute. A creature can repeat the saving throw at the end of each of its turns, ending the effect on itself on a success.

Cold Medicine: You can suppress any effect causing a target to be sickened. When this spell ends, any suppressed effect resumes, provided that its duration has not expired in the meantime.

Point being that your body is still working to fight off the infection, even if you don't see the symptom.