I have some scientific background (a BS in chemistry/biochemistry with quite a bit of emphasis on Molecular Biology and genetics), and as far as I know and have encountered, there is no real evidence of anything greater than lateral evolution.
I'm glad to see you can appreciate Rufus' comments. They strongly refute the premises of the opening post of this thread, using basic facts. (Great post Ruf!)
As to evidence of anything "greater" than "lateral" evolution (though you have defined neither term well),
reptile(synapsid) - mammal transitional fossils:
http://talkorigins.org/faqs/faq-transitional/part1b.html#mamm
You might want to also read this article. It approaches many of the separate lines of evidence for common descent:
http://talkorigins.org/faqs/comdesc/
After you read those two, you won't be thinking there is a lack of evidence for all but "lateral" evolution, unless you have a strange idea of "lateral" indeed!
If you go back far enough, the second law does seem to hinder the arguments of evolutioninsts and atheists. In order for the universe to ever begin there must have been something "outside" that dumped in the energy required to overcome entropy.
Of course, none of this is even remotely relevant to the issue of evolution, but I still would like to point out that you are assuming that the universe didn't start with the same amount of energy that exists within it now. If it did, then there was never any violation of the 2LoT. (That's second law of thermodynamics, not Lord of the Rings 2: the Two Towers

)
Ultimately, a hypothesis must still be made based on scant evidence and a whole lot of assumptions.
You have accurately described the state of most hypotheses at their inception. Indeed, you describe the state of evolutionary theory during the time of Darwin's musings on the Beagle.
Still, over time the theory has been confirmed by overwhelming evidence, though it had to be modified on some relatively minor points.
In my mind, belief in evolution requires faith in human logic just as much as belief in creationism requires faith in God.
All science requires faith in human logic. Without human logic you cannot use evidence to evaluate an hypothesis. For instance, with logic:
Hypothesis: water is a compound.
Logical statement: If a) water can be separated into more than one element, then b) water is a compound.
Test: Electolysis successfully separates water into Hydrogen and Oxygen.
Conclusion: Premise a) is true, therefore, premise b) must be true.
Without logic:
Hypothesis: Water is a compound
Test: Electrolysis successfully separates water into Hydrogen and Oxygen.
Conclusion: Electrolysis might successfully separate water into peanuts.
and I just wish that the schools would not teach the scientific/intellectual beliefs to children as if they were undeniable facts.
Does this go for gravity and other "deniable" facts that science has to offer, or is this a case of special pleading against evolution?