Accumulating evidence indicates that intestinal microflora
has protective, metabolic, trophic and immunological functions and is
able to establish a "cross-talk" with the immune component of mucosal
immunity, comprising cellular and soluble elements. When one or more
steps in this fine interaction fail, autoimmune or auto-inflammatory
diseases may occur. Furthermore, it results from the data that
probiotics, used for the treatment of the diseases caused by the
dysregulation of the immune system, can have a beneficial effect by
Gut microbiota interacts with both
innate and adaptive immune system, playing a pivotal role in
maintenance and disruption of gut immune quiescence. A cross talk
between the mucosal immune system and endogenous microflora favours a
mutual growth, survival and inflammatory control of the intestinal
ecosystem. Based on these evidences, probiotics can be used as an
ecological therapy in the treatment of immune diseases.
Antibiotics are designed to either kill or stop the growth of bacteria by targeting specific traits/functions of the bacteria (bacterial cells are prokaryotes) and not to target eukaryotic cells (human cells), which is rather easy due to the many differences between eukaryotes and prokaryotes. Some antibiotics target the cell wall of a bacteria (eukaryotes do not have a cell wall) or interfere with pivotal enzymes that allow them to survive and differentiate. Hence, antibiotics target prokaryotic cells and not eukaryotic cells, meaning that no, antibiotics do not weaken the immune system (this includes leukocytes (leukocytes, erythrocytes, etc. are all eukaryotes).
That said, antibiotics do not differentiate between harmful and non harmful bacteria, thus killing both indiscriminately. This means that the bacteria that were helping with normal bodily processes are killed, which may lead to problems.
The absence of functional/good bacteria in your e.g. gut and elsewhere also leaves you vulnerable to repopulation by unwanted/bad bacteria, which can be responsible for further issues.
But I would like to sound one note of warning. Penicillin is to all intents and purposes non-poisonous so there is no need to worry about giving an overdose and poisoning the patient. There may be a danger, though, in underdosage. It is not difficult to make microbes resistant to penicillin in the laboratory by exposing them to concentrations not sufficient to kill them, and the same thing has occasionally happened in the body.
I'll look at the first claim later. The second claim is becoming reality. There are a number of extremely dangerous bacteria that are now immune to most, if not all, antibiotics. The World Health Organization, along with many other organizations, is worried that Fleming's concern has become a severe problem. We may well be entering the post-antibiotic age. In the end, everything we do to defeat the immense powers of evolution are best short-lived solutions.
The problem is that we haven't discovered any new classes of antibiotics for thirty years. Thirty calendar years in terms of a dog's life is a long time. In terms of a bacteria's life, it's a long, long time. We may well be entering an age where a paperclip cut can kill. Those "natural" sites do not recognize that. Paperclip cuts could kill are 70 years ago, but not now. Those days will appear again.
Let's look at that first claim. While antibiotics are generally much more effective against gram positive than gram negative bacteria, using antibiotics on a massive scale kills almost kill all bacteria. Our gut bacteria are mostly gram negative.
Recent discoveries show that our immune system apparently depends,at least in part, on the presence of a healthy gut bacteria. Fleming's statement that "penicillin is to all intents and purposes non-poisonous so there is no need to worry about giving an overdose and poisoning the patient" is not correct.
However, it's important to remember Isaac Asimov's distinction between "wrong" and "not even wrong". The naturalists who reject antibiotics (and immunizations, which the linked articles reject) are "not even wrong." They are, IMNHO, worse than not even wrong. They are evil.