No Order of Operations Exists...
These are rules you'll have write. It's unfortunate in d20 that there's no quantified step-by-step order for combat and applying effects a la most trading card games. Were there, we could just say, "Immunity applies during step X, Damage Resolution, after Saving Throws but before Inflicting Effects," or whatever.
...But If You Want My Opinion
Yes, if only so the creature can choose to give up that saving throw.
Immunity shouldn't remove options everyone already has that aren't part of being immune. As everyone has the option of giving up a saving throw before knowing an effect's effect, immunity shouldn't change that.
Therefore when a creature is subject to an effect that requires a saving throw, the creature can choose to either make the saving throw or voluntarily give up (i.e. fail) that saving throw. Then the DM determines if the creature's immune to the effect.
Thus, unless the creature already knows an effect won't affect it (via a successful Spellcraft check, a successful Knowledge check that's revealed the opposition's abilities, or prior experience), the creature attempts the saving throw despite realizing an instant later that it possesses immunity to the effect.
That's because Pathfinder (and its antecedents) is a dangerous place, and what individuals can do varies wildly. A creature is safer if it always attempts saving throws versus affects, even if it thinks it might be immune to such effects because there's usually a random component to identifying effects, and being dumb gets it killed. Even a red dragon--who's immune to fire--will still make a saving throw versus the spell fireball [evoc] (Pathfinder Role-playing Game Core Rules 283) because there's no way to anticipate how a particular caster's fireball is going to differ from any other caster's fireball.
If immunity is checked first there's no opportunity to voluntarily give up the saving throw, and that opportunity should exist even if the creature's immune if for no other reason than to let the DM, when describing the spell's effect, tell the players that the creature appears to give up its saving throw, acting all awesome because the creature suspects he's immune.
If a Precedent's Absolutely Necessary
When spells are printed, they're printed with a Saving Throw first and the Spell Resistance after, so absent another order, those are checked in that order. Linking spell resistance to spell immunity is easy, but making the jump from spell immunity to immunity is harder, but, hey, the words are there.
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.
Best Answer
Yes
First we need to understand mechanically how immunity to a condition actually functions when a saving throw is involved. This can be observed by the following situation:
A Wizard casts Sunburst on an Ochre Jelly;
An Ochre Jelly is immune to the Blinded condition, but a saving throw is still called for.
The Ochre Jelly is even making the saving throw with disadvantage, but is still unable to be affected by the blinding effect of the spell due to its immunity, which indicates that a condition imposed by a spell as the result of a saving throw has no effect.
If a Paladin casts Thunderous Smite (PHB, p. 282) on the same Ochre Jelly;
This is another example of the one part of the effect of the spell simply being ignored while a saving throw is required to be rolled because the Ochre Jelly is immune to the condition that the spell inflicts.
This is all to say that immunity to a condition would just mean that if this condition were to affect it, it would do nothing instead.
The Fear spell reads as such:
The target still makes a saving throw as called for, but regardless of whether it fails or passes, it cannot be frightened. Any changes made by a portent or Cutting Words would be wasted if the characters were unaware of the immunity. However, RAW, this does also mean that a creature with immunity to the fear condition would still drop whatever it is holding as that is specifically not a part of the frightened condition, but an effect of the spell itself.
Saving Throws in the PHB pg. 179 reads as such:
And further in the PHB on pg. 205
Other than the slightly loose wording on page 179, nowhere does it indicate that a saving throw is ever optional or is not to be rolled regardless of the outcome of the effect, and a creature is unable to even willingly fail a saving throw as per Jeremy Crawford's response to this question.
Only the possible reading of "can" from page 205 would indicate a possibility of having an option of a saving throw, but this can be negated by the specific wordings of spells, such as the following:
A better example to consider for this question that has no extra clauses in the effect would be Blindness/Deafness
RAW, "to make a Constitution saving throw" indicates that regardless of the Ochre Jelly's immunities, the save is rolled. It does not state "unless the target is immune" or "if the condition imposed by this spell would have no effect on a success". On a fail, the condition simply does not take effect because of the immunities of the target. Additionally, this would also indicate that the spell is continuing for the duration or until a successful save is rolled.
Obviously as has been stated before, a GM may decide to rule it however they wish in the interest of time or enjoyment as they are the final word at their particular table, and I am sure that many would agree it would make the most sense to not make needless rolls or have your players waste resources; but as everything is written and strictly speaking a roll would need to be made regardless of immunity to the effect of a spell that calls for a saving throw because it never says otherwise.