Basic Area of Effect Rules
As far as Area of Effect goes...
A spell's description specifies its area of effect, which typically has one of five different shapes: cone, cube, cylinder, line, or sphere. Every area of effect has a point of origin, a location from which the spell's energy erupts. The rules for each shape specify how you position its point of origin. Typically, a point of origin is a point in space, but some spells have an area whose origin is a creature or an object.
The linked section goes on to explain exactly what each of the five basic shapes means, and none of them allow for shaping.
Shaping
One of the design precepts of 5th Edition is plain language - in other words, spells, features, and traits only do what they say they do. Take these spell descriptions for example...
You create a wall of fire on a solid surface within range. You can make the wall up to 60 feet long, 20 feet high, and 1 foot thick, or a ringed wall up to 20 feet in diameter, 20 feet high, and 1 foot thick.
An invisible wall of force springs into existence at a point you choose within range. The wall appears in any orientation you choose, as a horizontal or vertical barrier or at an angle. It can be free floating or resting on a solid surface. You can form it into a hemispherical dome or a sphere with a radius of up to 10 feet, or you can shape a flat surface made up of ten 10-foot-by-10-foot panels. Each panel must be contiguous with another panel. In any form, the wall is 1/4 inch thick.
You create a wall of ice on a solid surface within range. You can form it into a hemispherical dome or a sphere with a radius of up to 10 feet, or you can shape a flat surface made up of ten 10-foot-square panels. Each panel must be contiguous with another panel.
A shimmering wall of bright light appears at a point you choose within range. The wall appears in any orientation you choose: horizontally, vertically, or diagonally. It can be free floating, or it can rest on a solid surface. The wall can be up to 60 feet long, 10 feet high, and 5 feet thick.
A nonmagical wall of solid stone springs into existence at a point you choose within range. The wall is 6 inches thick and is composed of ten 10-foot- by-10-foot panels. Each panel must be contiguous with at least one other panel. Alternatively, you can create 10-foot-by-20-foot panels that are only 3 inches thick. [...] The wall can have any shape you desire, though it can't occupy the same space as a creature or object. The wall doesn't need to be vertical or rest on any firm foundation. It must, however, merge with and be solidly supported by existing stone. Thus, you can use this spell to bridge a chasm or create a ramp.
These spells all create walls but have highly specific and different rules for how those walls can be sculpted. Two can float in the air at whatever angle you like (Light and Force), the rest must be touching the ground or a particular material. Some are constructed with panels of a particular size (Force, Ice, Stone), while others have a set size (Light and Fire). Some must be a flat plane or a sphere (Force and Ice), while others infinitely shapeable (Stone) or can do cylindrical rings (Fire).
It's pretty clear from just these examples that the rules for shaping a spell, when available, are specific to the individual spell. What holds true for one spell does not hold true for others.
Conclusion
One of the core concepts of 5E is exception based design. If something is supposed to be different from the general rules, it will say so - and the explicit exception wins. The general rules are covered in Area of Effect, and make no provision for shaping.
As mentioned earlier, another core concept is use natural language. If it doesn't say it can be shaped, it cannot be shaped.
Specific Example
The specific example, Alarm...
Choose a door, a window, or an area within range that is no larger than a 20-foot cube.
One proposed answer to the linked question suggests you can break it up in blocks. The spell does not say a certain number of cubic feet, or panels, or a ring, or anything else. A room is an "area" in plain language, so Alarm fill a room's specific shape, as long as that room is "no larger than a 20-foot cube". If you're outside, it could cover a smaller area (like a clearing) as long as that area is "no larger than a 20-foot cube". But turning it into a ring (squared or otherwise) - nope.
Magic Mouth should trigger, but not Alarm
As always: many things are ultimately up to the DM, but the term "visual or audible conditions" in Magic Mouth implies only a visual/audible 'sensor'. Illusions should be able to fool visual/audible sensors, while a silent/invisible aberration should not trigger the Magic Mouth (by RAW it did not meet the visual/audible conditions). Giving the Magic Mouth spell greater detection abilities than this would seem unwarranted, and could lead to unexpected uses or abuses.
Conversely, the sensor for the Alarm spell somehow detects when a "Tiny or larger creature touches or enters the warded area" without mentioning sight/hearing. I'd use strict RAW here: an illusion wouldn't fulfill the triggering condition, but an invisible creature would.
For other spells - I'd similarly base the trigger detection abilities off their description. Examples:
- Detect Magic: "see a faint aura around any visible creature or object" implies that you wouldn't see it around an illusion (per Jeremy Crawford), but the trigger detection should still be fooled by Invisibility (an illusion spell) due to the "visible" clause in the above statement.
- Glyph of Warding can activate "under certain circumstances or according to physical characteristics (such as height or weight), creature kind (for example, the ward could be set to affect aberrations or drow), or alignment". So we know that it can detect all those things. But having it detect say, 'someone that lied in the last 5 minutes' (to serve as a lie detection spell) wouldn't seem warranted.
Best Answer
You can cast multiple instances of alarm
The general rules say that if you know the spell, have the slots, have the components, and meet any additional requirements, you can cast a spell.
You can even ignore the spell slots if you have the ritual casting feature and take 10 minutes to cast the spell as a ritual.
Casting alarm multiple times does not violate any of those rules and it is not a concentration spell.
Spells use language to describe effects outside of the rules
If there is not a general rule that describes something, then the spell's effect must say it for that thing to be true. A spell says what it does. Nothing more or less.
Other spells have language that specifically says what happens if you cast it multiple times. For example, minor illusion:
This gives credence to the interpretation that there are no rules for casting spells multiple times since it had to be elaborated in spell descriptions. But also, it makes the lack of such text in alarm significant and meaningful.
Prestidigitation even specifically allows, but limits the recasting of the spell:
Presumably it was important to put that limitation in there because prestidigitation is a cantrip and thus has no limit to the number of times it can be cast (unlike alarm which is limited by slots).
Similar spell, also able to be cast multiple times
An example of another spell that can be cast multiple times like alarm would be glyph of warding. It is similar in many respects to alarm and also has no language restricting it to one use.
Alarm does not have any such language and no general rule says you cannot do it, so you can!
You cannot increase the size of the spell effect by casting at higher levels
The spell says (as you also noted):
The spell will always specify if it has any effects that will scale with level or any other kind of details. The spell does not say anything about effects changing when you cast at higher levels so they do not change.
Ways to increase the effectiveness of your alarms
Using alarm in the middle of an open field and centering it on you while you sleep is not going to be the most helpful if someone is going to attack you for example. However, it is not useless and actually still could be helpful if the intent of that person was, say, to pick your pocket while asleep.
The clearest way to improve your use of alarm would be:
If you cannot find an enclosed campsite you could try any of these:
You can certainly cast alarm as many times as you want (make sure you are observing the casting time though), but really just picking a more secure campsite is going to be the best way to get better use out of alarm.
And it is important to note that alarm is a low level spell, it is not intended to solve all of your security needs. But in the right situation and with the right preparation it is invaluable.