Now - I cannot show that it is improbable but have the evolutionists have shown that random mutation ALONE IS responsible for these examples? Could they not be another unexplained force behind this?
In science, there is ALWAYS room for new explanations. The falsifiability of scientific theories requires this. But for science to consider new explanations you have to (1) actually provide one and (2) provide evidence that your explanation is better than the mutation alone one.
Its like I can propose that it is more advantageous for a creature to have a longer tail and viola a mutation appears that gives this longer tail .. natural Selection ends up with more long tailed creature.. Now the creature needs to whip this tail for some reason (it would be advantageous in nature for some reason) and magically another mutation gives it a stronger more flexible tail ...Is this the way evolution works? Anything can just happen magically?
What you seem to be suggesting is that natural selection would have some kind of foresight as to what might become advantageous. It doesn't. Rather, certain individuals carrying certain mutations are more likely to leave more offspring, leading to those mutations becoming more common in a population. There is nothing magic about mutations. Natural selection merely makes individuals with a certain mutation be more reproductively succesful, while individuals with other mutations are less reproductively successful.
Think about it like this - what are the all the possible errors due to mutation that could happen? And what is the likelihood that one of them will be beneficial? Has this been ever calculated mathematically?
There are certainly a good deal of "errors" that could happen. Some small (ie one base affected) and some large (thousands of bases inserted, deleted and altered, all in one go). In order to calculate the likelyhood of one being beneficial one would probably first have to determine empirically the fraction of mutations that are positive and those that are negative. I've only seen something similar to this being done for simple point-mutations, which does not reflect what really happens in real life. As far as I'm aware, a realistic model for this has not been done so far. But hey, there are thousands of articles published daily(?), so who knows? If you find one that does go into detail about this, I'd certainly be keen to know.
That's a bit of a non-answer, I suppose. So the absense of any kind of research into that area would mean that any kind of discussion concerning it would be speculative.
Is there something about a mutation that would make it beneficial or harmful other than the filter?
I would say that in some cases it is possible to predict with reasonable accuracy whether or not a certain mutation is beneficial/harmful. Since we know that certain genes are absolutely vital for survival, mutations that render these genes non-functional will by necessity be harmful (to conceive of such a mutation is easy - just delete the entire gene). To predict if a mutation would be beneficial might be more difficult, but given that an often given definition of a mutations is something like "A change in a DNA sequence", I'd like to give an example of a mutation that is highly likely to be beneficial in a certain environment: it is well known that bacteria often acquire resistance to antibiotics via horizontal gene transfer. Thus, a bacterium that has acquired an antibiotic resistance gene (which should fit the definition of a change in DNA sequence) would most likely have acquired a beneficial mutation (assuming that said antibiotic is common in it's environment and that the bacterium was not already resistant).
What I am perhaps confused about it that we have had folk claim that mutations are "harmful or beneficial or neutral" and that then go on to assert that most mutaions are harmful.
Studies have been performed to asses the impact mutations have on the fitness of organisms. Relative to wild-type individuals, fitness in mutants is almost invariably lower. This sort of makes sense, since there are more ways for a protein not to work then there is for it to work (all this assumes that the mutation happens in a sequence that codes for a protein in the first place). Experiments such as these do support the assertion that most mutations are harmful.
The problem with these experiments, as we both would probably agree, is the one of that filter you mentioned. These mutations were harmful in the environment in which they were selected for. They might not have been in another environment. Saying that, even if the environment changed, it would most likely still be true that most mutations are harmful.
Now the main problem is that we have not observed ANY such (similar)mutation in higher order animals such as mammals, birds, reptiles etc. That is a mutation that results in a physical change in the animal - a change in the function of a particular limb or organ. (And I dont' mean changes like long to short legs or size differences) - I mean something extraordinary - of this magnitutde - a scale becoming a feather or change in the animal's locomotion ...
If evolution was true, we would not expect to observe extraordinary changes (such as you seem to use the term here). While some mutations might produce some wicked instant new features (such as hens sprouting teeth) this is because the pathways involved in teeth formation are usually swiched off (and also revealing that the ancestors of hens had teeth). Insisting that the observation of extraordinary new feaures have not been done and concluding that this somehow falsifies evolution (if that is what you are doing) is simply a strawman.