Elsa must have created another "personal snow cloud", like she previously had for Olaf, for the cake. It's an ice cream cake (specifically mentioned), left outside in the summer heat, for hours (from Anna's waking up in the morning, until Elsa and Anna arrive at the cake at end afternoon - the tower clock showed 4:30 PM when they got at the tower, so it would be about 5 PM when they arrive at the cake). It should have melted in the heat by the time they cut it. How come it didn't and was still fresh? Well, it must have been kept cold by Elsa's personal snow cloud (or otherwise somehow magically protected by Elsa's ice powers).
Why is Elsa so convinced that she can't catch colds? Because colds are caused by a virus spread through human contact. She's been isolated for most of her life, meaning she wasn't exposed to cold germs often and hasn't gotten many colds. Unfortunately, this also means her immune system isn't as strong as it should be because it never really got used. There's also the fact that this takes place in a time period where people still thought colds were caused by exposure to cold temperatures rather than viruses. Since she's An Ice Person and is therefore unaffected by the cold, it makes sense that she would think herself immune to colds since she's immune to what she thinks causes them.
The much-beloved "open door" policy of Arendelle's ruling family (reinstated by Elsa and Anna after the former's powers settled down) might seem a bit reckless to viewers old enough to think about it. (What if a foreign power decided to assassinate the reigning monarch or her heir, for example?) However, any potential assassin would need to get through Elsa's powers, which not only allows her to generate blades from nothing, but also to generate an army of bodyguards and to freeze any attacker into solid ice. And that's for attacks against her - woe betide the country that succeeds in killing Anna, as Arendelle's Endless Winter would be but a taster of the terrors unleashed by a snow queen with family to avenge.
Elsa becoming increasingly inebriated as a result of (implicitly) drinking a bottle of cold medicine makes a lot more sense when you realize that, back then, many kinds of medicine contained a ton of alcohol. Even better, some of them contained opiates.
Snowlems that Elsa makes seem to possess a mindset depending on her current state of mind. I.e., Olaf is created when she's just playing with her powers, thus he possesses a childlike naivety. Marshmallow is made when Elsa is upset and thus has the mindset of a bouncer. And the Snowgies are made when Elsa is busy thinking about the perfect birthday party for Anna, which is why they aim for the cake... either because Elsa subconsciously wants a slice of the cake once Anna gets it, or Elsa's wish to throw the perfect birthday party for Anna causes the snowgies to try to steal the cake and bring it straight to Anna.
Remember the "We finish each other's..." "Sandwiches!" gag from the feature film? In this one, Anna really does finish a loved one's sentence, sleepily mumbling "..to you" when Elsa first wishes her "happy birthday". A subtle Call-Back to who really loved whom at the previous adventure's climax.
The giant snowball unerringly zeroes in on Hans from another country. The snowgies all converge on the courtyard where Anna's party is to be held without getting lost. Apparently, Elsa's unconscious creations have a built-in homing mechanism! Small wonder, then, that Olaf could find Anna so easily in the middle of the wilderness, or make his way through a large and unfamiliar palace to the room where Hans had locked her in to die: Elsa'd created Olaf from the memory of her childhood playtime with Anna, and unknowingly granted him the ability to always find her little sister.
The snowgies also, without being told, involve themselves in surprising Anna in the courtyard. And Marshmallow is this furious manifestation of Elsa's fear when first created, but apparently a perfectly pleasant fellow once Elsa herself calms down. Maybe it's more than a homing mechanism; Elsa's creations are independent-minded, but also subconscious extensions of her thoughts and her psyche.
At the beginning of "Making Today a Perfect Day", Elsa says she's done everything within her power to make Anna's celebration a success. In the end, when she thinks it's been "ruined," the thing she believes did so is something Elsa had no power over.
Elsa wearing green — In Frozen Fever, Elsa magically changes her signature light blue dress into bright green which has several levels of meaning. 1) Elsa changing her clothing into Anna's color represents her efforts in this short to open up her heart to Anna and let her in (after all, it is all about her giving Anna a perfect birthday to make up for the previous birthdays she missed out on). Plus, if you're familiar with the symbolism of the Chakras: green stands for the Heart Chakra or love - in this case sisterly love. 2) Green being the color of spring or summer more than winter represents Elsa not being overpowered by her ice powers (which can be represented as blue) anymore, but being in control of them and NOT being in danger of changing it into a winter anymore.
Anna and Elsa never notice the snowgies running around until after the "SURPRISE!" in the courtyard, so Anna ends up even more impressed than Elsa intended; she would think that the pile of tiny snowmen and fireworks effect was part of Elsa's plans all along.
Elsa's ice creations tend to be manifestations of her subconscious. It seems a little odd that Elsa would create an obviously inappropriate ice sculpture (one of her weeping over a frozen-solid Anna) as a cake topper, that she would never consciously include. However: the birthday party - and particularly the cake - are intended to help make up for all the years Elsa neglected Anna. Elsa is subconsciously running through the aspects of her past relationship with Anna that she wants to make up for, so the sculptures she creates are all representations of that, and in order too. First is Anna, alone, as she was outside Elsa's door. Second is "stiff", as Elsa puts it: formal and distant, as at the ball. Third is Anna's sacrifice. Elsa feels a little guilty about each one and that is manifesting itself in her attempts at sculpting a cake topper. Eventually her relationship with Anna progressed toward the sisterly togetherness that they both wanted, and that's what appears at the end, in the form of the two ice skating together.
Throughout the entire series, Olaf is shown to be (somewhat) intelligent and articulate, while Marshmallow only speaks in simple, broken English and the Snowgies don't seem to speak at all. Why would Olaf's siblings be less developed than him, even though they were all brought to life by Elsa? Because with Marshmallow and the Snowgies, Elsa made them both on a whim - and in the latter's case, completely by accident - without putting any thought into them. Whereas Anna and Elsa first built Olaf from scratch when they were children, they imagined a clear and distinct personality for him ("I'm Olaf, and I like warm hugs") and Elsa clearly had this memory on her mind when she rebuilt him during Let It Go. In other words, Olaf is literally a fully-formed character.
That scene where the giant snowball hit Hans was a cute little throwaway joke, right? Well, imagine that you are the ruler of the Southern Isles. You sentenced Hans to hard labor mucking out the stables, and you think you have done justice while appeasing Arendelle's anger. Now, suddenly, a giant snowball flies across the sky and scores a direct hit on Hans. We know that this was just random chance, but you, as the ruler of the Southern Isles would never think this. You would first conclude that this was a sign of Elsa's displeasure. You would then conclude this must mean that Elsa's powers include the ability to scry or remote-view what is happening at great distances and the ability to see at least some time into the future, since there would be no way to score a direct hit at such distances unless you could not only see where Hans was at the moment when you launched the snowball, but also where he would be by the time he arrived. You already know that Elsa can destroy a nation in her wrath. Now you think that she is, if not omniscient, then at least possessed of supernatural knowledge that makes her more dangerous. You will of course increase Hans' punishment, despite the fact that increasing a sentence that has already been handed down is a clear violation of the rule of law. What else can you do? Think how much terror that little throwaway joke will provoke in the Southern Isles and beyond if anyone else finds out about it.
If anyone saw the snowball, that is. It'll melt pretty fast, and it moved pretty fast, so it's entirely possible the only person that knows anything about a giant laser-guided snowball is Hans. Which is a different twist on the same Fridge Horror. People would wonder why he's covered head-to-toe in horse poop and might wonder if he hadn't been rolling around in it. If they ask, and he says "no, really, it was a giant snowball headed straight for me", they'll think he's crazy. So he'll just say "I fell," and keep the snowball thing to himself. Which means it won't be the ruler of the Southern Isles wondering what to do. It'd just be Hans, believing Elsa is still furious at him and that at any minute, giant snowballs or worse (say, icicle spikes or another big angry Marshmallow, both of which he's personally seen) could come tumbling out of the sky, straight at him, wherever he happens to be.
Elsa says she doesn't get colds, but clearly she does. She probably thinks she doesn't because she was holed up in the castle for 13 years and wasn't exposed to many germs. Now that the gates are open (and she's no longer wearing gloves, for that matter), she's probably going to get sick far more often. Unless she learns to control her sneeze-induced magic, Arendelle is going to be flooded with Snowgies for years to come as her immune system gets up to speed.
They take all the Snowgies to the ice castle where Marshmallow lives at the end of the short, nothing stopping them from doing that each time Elsa gets sick. How Marshmallow feels about this is another story.
Still, the snowgies are shown to be mischievous creatures, and if Elsa is creating them unknowingly, they could cause a lot of trouble.
If Elsa didn't take the "medicine", could the reason for her being more delirious than with a normal cold be that she has a very weak immune system? And if so, that means that she's more in danger of dying from disease.
