> 3) What is A∨B, using the definition given in the paragraph above

I am choosing A and B as dented on the picture below

Ex3_Chapter1


So
A∨B is a subset that has a union of blocks in A and B (hope this wording is correct).

Therefore, it is

Ex3_Chapter1_p2


> 4) Is it true that A≤(A∨B) and B≤(A∨B) ?


I think this is true.
The effects of the ∨-operation are 'additive' and cannot be destructive. And, because ≤ - relation is defined as 'whatever is connected in A, should remain connected in (A∨B) , we cannot have a situation where there is a connection in A, but not in (A∨B).


> 5) What are all the partitions C for which both A≤C and B≤C


In my case (because of how I chose A and B), it is


Ex3_Chapter1_p2


> 6) Is it true that in each case, (A∨B)≤C ?


It is not true in general. Although this happened to be true in my selection example.
It is not true in general, because ∨ can produce generative effects, such as new connections, that were not part of C , given how C could be selected according to 5).