A lot of good introductory questions here - I can tell this is really bugging you. Not near my computer at the moment so no pictures but maybe we won’t need them. First off, we need to be clear on some definitions. I’ve paraphrased a lot of your questions from above and hopefully all will be made clear.
What is a “beat”?
This is how I explain beats to 5yr-olds, and if they get it, you should be fine. I have them put their hand on their chest and feel their heart. I tell them that their heartbeat keeps them moving. Sometimes it’s fast, sometimes slow, but it’s always steady. I tell them that music also has a heartbeat: sometimes fast, sometimes slow, but always steady. In music, we call the heartbeat the “pulse” - it’s the thing that you feel, just like when you feel your heartbeat. (Pulse and beat are interchangeable, really, though pulse is used to describe beats in a more general, non-specific sense.) in music, we show the pulse with a symbol called a “quarter note”. Each “quarter note” represents 1 pulse. So, if I have 5 quarters, I have 5 pulses or beats. If I have 1000, then 1000.
At this point I have a metronome going and I show them the numbers - that a metronome @ 60 will make exactly 60 clicks in one minute - each click representing a beat. Then I change the numbers fast and slower so they can hear the result of the number changing. Then I play the piano, showing them that for every quarter note I see in the music, I make exactly 1 sound that lines up exactly with the metronome click. (At this point we’ve already practiced with just lining up with the metronome by tapping on our laps).
There you have it, a beat.
What is “meter”?
Meter is how we organize the beat. If we don’t have meter, it’s hard to read the music and play with other people. The most common meter is 4/4 time in which there are 4 beats per measure and the quarter note gets the beat.
What is a “measure”?
A measure is the space between bar lines. I tell my students to think of it like an empty box and the beats like blocks. If we have 4/4 time, we can only put 4 blocks in the box (if each block is worth 1). 5 blocks can’t go into 4/4 because it’s too many. However if we made the box bigger (5/4) then our blocks would fit just fine.
Why can you fit 8 notes in 4/4?
The answer here builds off the last question. You can think of it like blocks, but I prefer a pie chart for explaining 8th notes. This answer may also be helpful to you Europeans who wonder why silly Americans call the notes like we do.
First, draw a circle. This represents a whole measure in 4/4. It also represents a whole note, which is 4 beats long and is called a whole note because it takes up the whole measure.
Now divide the circle in half. These new pieces are called half notes. They are each half as big, but when you add them together, they make a whole measure. If a whole note is worth 4 beats, then a half note is worth 2.
Now divide the circle again, the other way, to yield 4 equal parts. These are called “quarters” because each one takes up one quarter of the measure. 2 quarters equal a half, and 4 a whole. Each quarter is worth 1.
Now divide each quarter. We now have 8 pieces in our pie. Since 2 8ths fit into 1 quarter, we know that each 8th must equal half of a beat. They’re called 8th-notes because 8 of them fit into a whole note.
You can keep on dividing them smaller and smaller though I don’t believe I’ve personally seen smaller than 256th notes.
Can a beat exist on its own?
Yes. You can describe how a particular rhythm divides a single pulse.
Is there more than one way to show a beat?
Yes. This is the reason for the lower number in a time signature. A beat may be represented by any note. In the case that the lower number is not 4, then all other rhythms behave proportionally to that number. For example, if the time sig is 6/8, the 8th carries the pulse. That means that a quarter note is now worth 2 beats and effectively functions like a half note. 16ths would function like 8ths as they now get half a beat each.
And yes, you can divide a whole note by any number and that number can get a beat. So if you wanted to divide by 3, you could have 3/3 time, or 4/3 or 4/5 or 7/11 time. No, it’s not irrational like many people claim. There’s nothing irrational about the numbers 3, 5, or 11. It’s just uncommon.
Is a beat the same from piece to piece?
For music that uses beats, yes, the basic concepts are the same - whether the piece is slow or fast, the meter asymmetric, or the pulse represented by an uncommon number, the core concepts I shared here today remain.
Welcome to the site, and hope this clears it up for you.