Is there a temperature difference from a ground floor apartment to a top floor apartment, in a 30-story building?

Yes and no.

If you had a simple structure 300 feet tall and completely hollow. The thermal energy from the exterior (sunlight, air temperature, …) would heat the column of air and distribute it so that the heated air would move upwards leaving the cooler air below. Depending on humidity and thermal conductivity of the structure would determine what happens to the air at the top (rain, cooling, …).

But I think you are referring to a modern 30 story building.

We go to great lengths to environmentally control each part of the building to the level desired by the occupant, so the temperature can be arbitrary within human comfort temperatures.

So, is it hotter or colder at any level; depends on the occupants. If there more thermal energy present at certain areas such as the top versus the bottom; yes.

What is the lowest part of a building?

There are two ways to look at this question. What is the lowest part of the structure, or what is the lowest human occupied part of the building.

First question. The answer is the foundation. this is the part of the building which interfaces with the building and ground it sits on.

Second question. There is no special or specific answer for this. Typically, it would be the ground or first floor as it sits directly on the ground. Rooms that go below the ground (at least by a significant portion) are called basements. Most of these rooms are named by use rather than by location.

Why does putting a heavy book on a crumbled piece of paper make it flatten out, and why does it take 10-15 minutes for it to happen, not instantaneously?

Okay, the answer to this can be rather complex.

First, don’t think of the paper as a single object but rather as a net (really more like threads placed in random directions) of single thread-like fibers. When you crumpled the paper you bent each of the threads along each crease beyond their ability to bend (think of metal bars that are now bent). Each of the fibers is either broken, cracked or deformed (narrowed along the length of the bend).

Now you flatten it out. The broken and deformed pieces no longer fit the same way as the original flat paper as they no longer have the same shape. Now these fibers are pressed against each other (from the book) and must drag themselves across each other until they reach a point of equilibrium between the pressing of the book, the friction of the other fibers and elasticity of the fiber itself (the tendency of the fiber to assume its neutral shape).

Just like things at our scale need time to move and shift, this is also required at the scale of the fibers as well.

If you want a slightly more mathematical answer we can go with Newton’s Second Law (F=ma). If you apply a force to any given mass, a time component is needed as part of the movement. This continues until all forces come into equilibrium.

Why do people build their homes out of concrete? What are the advantages?

The answer comes down to any one of several points.

First, is it readily available as a building material? Throughout most of the world there are two favored building materials at the single home scale. Wood/timber and concrete/masonry. Basically, if wood is scarce or not suitable due humidity, mold, or infestation concrete or clay blocks, masonry, becomes the preferred method.

Second, strength or safety of the material. If you live where natural occurrences knockdown or destroy homes, like fires, occur regularly concrete or masonry construction tends to be more resistant, even without engineering. If, on the other hand, you are trying to build a large structure (spread out and/or tall), concrete is one of two options to obtain the strength and fire resistance you will likely need.

Last is aesthetics. Given that concrete is a very strong material when compared to wood while not being particularly confined to given shapes, it is a preferred material in architecture. It can be sculpted, shaped, textured, colored as one sees fit. You can also use smaller structural systems when compared to typical systems. (I do understand that concrete weight tremendously more than wood which results in thicker, more robust structural systems.)

What is it like to live in a basement?

Let’s break this into two segments; you’ll see why.

First, let’s work on a basement intended for human living.

Generally, it will be cooler than most similar spaces due to the extra insulation (dirt) outside, unless it is below freezing outside in which case it will tend towards a little warmer. Most of the natural light will occur in the upper third of the wall if not from almost skylight like light wells. The ceiling may be a bit lower, and you’re likely to see a lot more of the large beams projecting out of the ceiling. The walls will tend to be a bit more irregular in size and shape than you’d be used to, along with the ceiling and portions of the floor. Columns would be common. Otherwise, similar to other living spaces.

Keep in mind, if there are building problems with water, sewer, HVAC, whatever… you are likely to have them first and more severe.

Basements intended for storage.

Take the situation above, but there will be no sunlight or windows of any kind. The space may not be heated, cooled or even ventilated. Ceiling height may become very low with exposed building systems. The floor may be finished but may end to dirt abruptly. If there is exposed dirt anywhere dampness is likely to be an ever-present issue.

Basically, you would be living in the modern equivalent of a cave.

Can you make a mailbox post out of cement as a deterrent for cars hitting it?

Basically, you are looking to install a crash bollard with a mailbox mounted on top.

This is a detail of one. Typically, it is restrained at the ground by pavement, but isn’t required. The bollard itself is a 4″ diameter schedule 40 metal pipe or 4″ A36 steel pipe. Some designers also add rebar (4-#4 bar vertical with #3 bar hoops at 12″ on center). (#4 bars would be 2′-0″ each and hoop would be 1′-0″ diameter for this detail.)

Keep in mind, this is not crashproof, but it is rated to prevent a large vehicle (think loaded semi or trash truck) from moving past it.

Are all buildings required to have a sign showing the maximum occupancy?

Simple answer; no.

For the most part, only rooms where people are expected to gather for some length of time. This number doesn’t represent the actual amount of people that can be packed into a space, but the amount of people in the space at a predefined density. It is to be used as a guideline for fire safety as the space was designed to get at least that many people out of the space within limits set by national fire safety guides.