Learn English – until July 28th – meaning

This looks like an example of someone with ESL having difficulty with the use of the tense. Note the "until the last month" which is another typical error; "until last month" is correct. They both have the feeling of "foreign accent" to them.

"Have been in the middle of" means that the state of being began at some unspecified point and continues at present. "Were in the middle of" means that the state ended at some specific point in the past. "Had been in the middle of" means that the state was ongoing over a period of time in the past, but usually also involves some sort of specific or implied explanation of what caused the state to end. For example "We had been in the middle of a peace process when the bombings last month ended it."

In this case, I think the meaning is best expressed as "We are...we were in the middle of a peace process until last month."

It’s too late to go shopping. The shops are open only until 5:30. They will be closed by now.

Yes, closed here is employed as an adjective. You are correct in understanding it to designate a state rather than an event.

And it is true that the expression by TIME is used with events rather than states—as your source says, it says that something ‘happens’ (an event) rather than ‘continues’ (a state).

HOWEVER: The expression by TIME does not compel you to employ an ‘eventive’ predicate or forbid you to use a ‘stative’ predicate. In fact, the compulsion works in the opposite direction: when you use by TIME, you compel your hearer to understand the predicate in an eventive sense. With by TIME the hearer interprets a stative predicate as the result of a change of state: something happened before TIME to bring about the state you describe.

Here are examples of by TIME with three stative verbs, know, own, be:

By April he knew that the operation was a failure. ... implies that he did not know this earlier, but at some time before April he learned that the operation was a failure

By 1973 he owned 26 newspapers. ... implies that he did not own so many newspapers earlier, but at some time before 1973 he acquired 26 newspapers

By next week you will be in Toronto. ... implies that you are not in Toronto now, but at some time before next week you will go to Toronto

In the same way, The shops will be closed by now implies that earlier the shops were open, but at some time before now the shops were closed. Since it is explicitly stated that the shops are open until 5:30, we understand that this sentence must have been uttered at some time after 5:30.

The word will may have caused you some confusion. This will does not designate some time in the future; it expresses a certain inference in the present. (Linguists call this ‘epistemic’ will.) Your last sentence may be paraphrased

It is certain that they are closed by now.