you ignore all class, race and level requirements on the use of magic items

and

using your spell attack bonus

From p.207

Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.

A spellcasting ability modifier is only given as a specific class ability. Barring multi-classing, a 13th level thief does not have a spellcasting ability modifier so this is 0. They have a proficiency bonus of +5. So +5. (DMG p.141)

Similarly for the saving throw.

Give it to the wizard!

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.