You're a bit confused about the terminology: in order to show that HAMPATH is NP-complete, you need to show that it's in NP (that's very easy) and that every problem in NP reduces to it. You do that by reducing some other NP-complete problem to HAMPATH.
As Theo noted, 3SAT is a special case of SAT, and so it's easier use 3SAT. The reduction that works for SAT will also work for 3SAT, but it may need to handle more cases. I'm not familiar with this particular reduction, but it might be that it's actually straightforward to generalize it to the "general case", but there's nothing to be gained by doing it.
Reductions showing a problem is NP-complete are usually like that - very intricate and slightly hard to follow. But usually there's the main idea - e.g., what corresponds to variables, what corresponds to clauses, why it should work - and the technical details. See first that you understand the main idea.
The technical details break into two parts. One part is the gadgets - you need to understand what the gadgets are supposed to do and how they accomplish it. The second part is proving that the construction works. In your case, it's probably easy to show that if the 3SAT instance is satisfiable, then the graph has a Hamiltonian path. It's probably more difficult to show that if the instance is unsatisfiable, the graph doesn't have any Hamiltonian paths. There's some intuition which leads us to believe that to be the case, but we still have to prove it, and the proof might be tricky.
Don't get discouraged if you don't "get it" right away. The best thing to understand these proofs is to do some of your own. Only then can you appreciate the thought processes behind such proofs. Better still, you might try working out a similar proof yourself, and see where you get stuck. There are some obstacles to overcome, and if you're aware of them then the construction will become much more interesting.
Looking at the reduction, I think Sisper explains it pretty well. In fact, for an expert it might be enough to mostly look at all the figures. The pattern of this reduction is very standard, so once you understand this one, you'll be in much better position to understand a whole lot of them!
The idea is to have variable-gadgets and clause-gadgets. The variable-gadgets can be traversed in two ways (according to the truth assignment). The clause-gadgets can be traversed-through from a satisfying literal.
So far for the idea. The construction itself is pretty simple, and it's easy to check that if a formula is satisfiable then there's a Hamiltonian path. It's also easy to see that if there's a similarly-looking Hamiltonian path then there's a satisfying assignment. The hard part is showing that all Hamiltonian paths "look as if they came out from a truth assignment", which must be satisfying.
Take your formula and check if it is satisfiable. If so, conjoin $x_1$ to your formula, where $x_1$ is your first variable, and check if it is still satisfiable. If so, then there is an assignment where $x_1$ is true; otherwise, there is one where $x_1$ is false.
Conjoin $x_1$ or $\neg x_1$ and repeat for all of the other variables.
Best Answer
For the problem to be NP-complete it has to be in NP, which means it has to have a polynomial time verifier for all "yes" answers. If the exactly-1-satisfying-assignment question is in NP, that means there must be "yes" answers and verifiers for the following two questions:
Does this formula have a satisfying assignment? Does this formula have no more satisfying assignments?
There's a polynomial time verifier for the first question, but not the second unless NP = coNP. Since the status of NP = coNP is unknown, the status of the main question also has to be "unknown."