1. Philosophy isn't concerned with empirically testable theories
In philosophy, while new theories must comply with existing theories, there is clearly much less [empirical] testing/retesting.
This observation is quite correct (leaving aside the remark about new/existing theories, see below).
However, the claim that philosophy doesn't (shoudn't) deal with scientific (empirical/mathematical) questions and methods is rather new, compared with what philosophers have been actually doing for most of philosophy's history. This new claim appeared ca. 250 years ago, which is a rather short time period compared to philosophy's 2500 years of existence. Up to German idealism it was customary for philosophers to be not only deeply informed about scientific matters, but to engage in scientific questions as well.
(To be sure, there were always battles within philosophy between rationalistic and empiristic inclined philosophers, and so Descartes would contribute to mathematics, while Newton would contribute to physics. But all these philosophers wanted to advance, in different ways, our actual knowledge of the world.)
One important symptom in the modern self-(re)imaging of philosophy as non-empirical was a debate occured in the 19th century about whether psychology, which was at the time a subdiscipline of philosophy, should use properly experimental methods in research. The faction supporting this view could succeed only at the cost of leaving philosophy (institutionally) in order to become an experimental discipline. The so called Psychologismus-Streit (psychologism debate) which followed sanctioned that logic and, most importantly, epistemology shouldn't have anything to do with psychology, thus consolidating the claim that philosophy shouldn't employ empirical methods and should deal with conceptual issues only. This is pretty much the status quo today.
There are, however, current attempts to return to the older view, i.e. that philosophy should employ experimental methods (check out experimental philosophy).
2. Does philosophy meet scientific rigor?
In science: A theory must have observable consequences that can be tested and be falsified.
Theories also must agree with previous theories in the domains where they had been successfully tested.
For example, special relativity reduces to Newton mechanics in the domain of velocities much less than the speed of light.
In philosophy, while new theories must comply with existing theories, there is clearly much less testing/retesting.
You seem to imply that in order to be called a science, a discipline has to produce empirically testable theories. This certainly isn't an evident premise, as formal sciences do not meet this criterion. Are they therefore not scientific? This would seem an odd conclusion (and I don't think you would like to draw it).
(Results of formal sciences are clearly used in constructing and testing scientific models dealing with observable reality, but these sciences are certainly not tested themselves this way.)
So, how does philosophy fit into this framework while maintaining the rigor required for scientific inquiry?
Well, when philosophy aims to maintain a certain rigor, it does so by adopting instruments used in the formal sciences (such as logic, probability theory, theoretical linguistics, etc.). But some branches of philosophy do not actually aim to emulate the epistemic values found in science, they orient themselves more at epistemic values found in other fields, such as literature. But still, they are certainly bounded to common standards of academic inquiry.
(As to your characterization of scientific theories, see the answer of Michael Dorfman's reply. There is a common problem when speaking about "what science is" or "what scientific theories are", as it is not clear if one means to give a (widely shared) definition or a factual description. Most of the time it is meant both as a definition and a description. Historical and sociological research showed that some widely shared definitions are simply false when used as factual descriptions of science - which still doesn't invalidate these claims qua definitions!)
3. Do philosophical discussion influence scientific theories?
Are there any examples of philosophical discussions that have significantly influenced or changed scientific theory?
Yes!
As you might see from the above, formal and empirical research was actually a part of philosophy for more than 2000 years. So the answer is: philosophy itself!
But probably you mean "science" in the sense of contemporary science, i.e. as a clearly differentiated set of disciplines with no general institutional ties to philosophy. Even in this case the answer is still: Yes!
As you mentioned Special Relativity (SRT), I'll take this example to mention two cases in which philosophical discussions were involved in creating and defending/advancing SRT as well as General Relativity Theory (GRT).
Ernst Mach formulated repeatedly the (now so called) Mach's principle, which was seminal for Einstein to conceive GRT. (See this page for more info.)
Since the Lorentz ether theory (LET) and SRT were deemed to be empirically equivalent, most scientists were puzzled as to which criteria could be used to solve this problem of underdetermination. Physicist Max von Laue consulted on this matter a still young Moritz Schlick, later to become the founder of the Vienna Circle, logical empiricism and thus of most of modern philosophy of science. Schlick gave a new interpretation of the principle of simplicity, showing why on these grounds SRT should be preferred (the old interpretation of simplicity actually supported LET). Together with Schlick, other soon-to-be logical empiricists, such as Hans Reichenbach, joined the discussion and formulated the first interpretations of SRT. They were also really important in defending SRT against scientifically and not-so-scientifically minded attacks. (Follow this link for detailed info on their contributions to SRT.)
If you need more (or more impressive?) examples, please let me now in the comments.
