1. Sometimes I get confused about the uses of 'upon' and 'on' sentences. Do you have some tips?

contingent upon/on

dependent upon/on

2. What is the difference, using 'lay' and 'lay out'? Very little or no difference to me.

Please lay your paper in front of you.

Please lay out your paper in front of you
To be honest, I'm not sure if there IS much difference between upon and on in the examples you cite.

With "lay" it just means to set it down flat. "Lay out" has a sense of spreading it out. If I have a rolled up mat that I've brough to my yoga class, I can lay it down by putting it down, still rolled up. But if I lay it out, then I unroll it.

I would know say "lay out your paper" unless I was using it in the special sense of journalism, when you decide what will go where in the newspaper.
The most common usage of upon tends not to be in the sense of adornment like 'having something upon your head'. Usually it's:

- in fairy tales: "once upon a time.....",

- in taking on a task or action voluntarily: "he took it upon himself to challenge the decision.... "

- something forthcoming: "the exams will soon be upon us...."

Where the context is of having something on top of something else, then it tends be 'on' that is used rather than 'upon'.
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Upon is the archaic and formal variant of on, AFAIK.
 Tidus's reply was promoted to an answer.
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"Upon", in some uses (the last 2 in Tidus's examples) adds some movement to the rather static "on".
There is no longer much specific grammatical distinction between the two. Often, the prepositions "on" or "upon" can be used interchangeably; however, certain verbal phrases demand the use of one or the other specifically (I switched on a light, wherein "upon" can not be used; I came upon a fork in the road, wherein "on" can not be used). Verbal phrases are idiomatic and often open to more than one possibility.

Some background might help to understand why we have two prepositions often meaning the same thing. Earlier English used prepositions to relay a sense of movement from one place to another. Pieanne is exactly right.
For example, we can say:

The plane rolled on the runway.
The plane rolled upon the runway.

I walked in a room.
I walked into a room.

We drove between the buildings.
We drove in between the buildings.

In modern English, the sentences are mostly interchangeable; but the different prepositions were originally meant to relay a sense of change in surroundings. The second example makes the difference most evident: are you walking around in one room the whole time, or did you enter the room?

I would say that a fly lands upon a table, then the fly walks around on the table. Keep in mind there is no specific rule, and there are plenty of idiomatic phrases to go along with the prepositions.