This is a huge subject I'm not about to take on completely, but I've had it on my mind for some time since seeing a documentary about the Logical Positivists and reading some of Husserl's work.
Obvious questions:
In addition I would add these questions to muse on:
I am not interested in discussing the first three questions here. Philosophy and science are both legit. One is descriptive and the other applies to the understanding. I agree, some philosophy will stop being philosophy. But not all. This is due to the fact that some speculative physics is placed under philosophy or because some legitimate philosophy has made speculated about the underlying physics which may turn out to be incorrect. These other questions though are more fundamental and must be explored. Only through proper method and careful consideration would those three questions be answered.
So, is there a science of science? Isn't that a bit circular? Yes, but scientific method is a process for investigating phenomena correctly and rigorously and with authority. But science is not a phenomenon, it is a body of knowledge. So I'd argue that no, there is no science of science.
But what about philosophy of science? Of course you can argue the foundations, details, implications of any topic, but philosophy of science is interesting because it seems like all of metaphysics is nothing but a discussion revolving around knowledge, definition of, perception, truth and the ultimate reaches of same. Moving beyond that I think it is important to understand a) the attitude and b) the ethics of science and conducting science and also c) the establishment of methods that can produce a body of knowledge which explains phenomena with authority. Finally we must all d) agree on methods and e) revisit them and improve them. Beyond this discussion is topical. (Why we do this or anything is under the purview of philosophy but does not belong here.)
To conduct good philosophy you need method, process and tools. You need a method to communicate your ideas. You can't just write something down and explain a philosopher or scientist to believe it without backing it up. This is the science of philosophy. So here we are concerned with the development of tools and methods, mainly logic and other types of thinking, that enable us to do good philosophy. This is tricky because we must use thought and logic and imagination to produce tools with which to think and rationalize. It's a miracle that we can do this and make any progress, but we do. This is also a bit tricky because it can be argued that since there is nothing to physically observe there can be no science of it. But thought is observable in the mind and whether or not anything can be proved may not matter as much as everything else that can be disproved.
Lastly, what of philosophy of philosophy? This is not separate from philosophy, it is philosophy. It is the philosophy. It is a philosophy about all other philosophies. It is of course abstract, having philosophy as its only subject. But you could not seriously have a system of philosophy which is both thorough and correct without addressing this at some point. It asks why philosophize? What are the limits of philosophy? Where does it give ground to science and what does that intersection look like?