Yes, the first statement is a "natural" construction. And yes you can use the second statement instead. It wouldn't impact your formality in a big way, and it's not impolite, though all things being equal I guess you could say it's slightly less formal or polite because it contains slightly less polite hedging.
Asking "could you" in this context is not about ability, and "would you" is no better in a literal sense because it implies a hypothetical situation, which this is not. In the same way, the speaker's point is not really about his own "wondering." These are all ways to couch a request politely, and often the bigger the couch, the more polite the request.
Literal responses to these kinds of requests might look like this:
A: I was wondering if you know what time it is.
B: Oh. That's interesting... do you spend a lot of time wondering what I know?
A: Could you help me with this broken pipe?
B: Yes, I could.
A: Would you help me with this broken pipe?
B: If what?
We can build a progression of requests or questions that illustrates the effect of polite couching. As the expressions get longer, the actual thing being asked for gets buried deeper, becomes more indirect, and is placed further away in time and possibility from the speakers:
- What time is it?
- Do you know what time it is?
- Would you happen to know what time it is?
- I was wondering if you happened to know what time it is?
- I'm terribly sorry to bother you, but I was wondering if it would be at all possible for you to tell me what time it is.
Splitting hairs about the relative politeness of the original two statements in the post's question is not going to bear a lot of fruit, because they're so close that how well your shirt is ironed and hair is combed (not to mention your intonation and body language) will have a greater impact on formality than your choice between the two constructions.