I have stack-exchanged through my undergrad math program. Am I likely to succeed in mathematics PhD programs?

Yes! In fact, I think you're well on your way to doing better than your peers! Taking longer to understand something isn't something to be proud of! There's no need to reinvent the wheel. If someone can help you understand something, you would be well-advised to make use of them. In the same way, you would be well-advised to attend the lectures, thereby getting help from the professor, instead of staying at home with the ZFC axioms and attempting to derive the whole of mathematics from scratch.

My experience asking questions on Stack Exchange also indicates that simply writing the problem in a form which others can understand is a great help in clarifying my own understanding of the problem. Indeed, I've solved some of my problems simply by beginning to write a (never asked) question.

Of course, this doesn't mean you should outsource your understanding to others. You should make it so that you can explain the concept to next year's students without help (answer some other SE questions while you're at it!). But even then you should feel free to ask for help with next year's problems.

First of all, I believe this is extremely common these days. More and more I notice students neglecting to develop important problem solving skills and instead developing great “google-fu” and “stack-exchange-fu” skills to achieve the same goals.

Now, don’t get me wrong, SE-fu is a terrific skill to have. Just like you are worried about using the internet too much and coming to rely on it as a crutch, some people genuinely ought to worry about having the opposite problem of obstinately trying to figure everything out themselves even if it takes them weeks or months and refusing to ask for help. This is just hugely inefficient. These people may in fact become excellent problem-solvers given enough time, but it’s just not a practical approach to covering the large amount of material a modern mathematical education requires.

So what I’m saying is, there is a right amount of stack-exchange usage that can be really good for you. Someone who makes the right use of math.SE and other great online resources can really boost their ability to master complex topics and speed up the learning process compared to their peers who don’t use those resources. And then... there is definitely also a wrong amount of stack-exchange usage. It is certainly possible to rely on it too much, or more generally to rely too much on asking other people to help you figure things out when you get stuck (before SE was around, people with such tendencies also existed, they would just nag their friends and class mates with lots of questions instead of using google/SE).

So, is this a fatal flaw or an indication you’re unlikely to succeed? Absolutely not. From your description it sounds like a slightly bad habit at worst, but one that you likely share with a lot of other students (I mean probably something like 70% of them, if we interpret your habit to include heavy use of google and not just SE).

I do advise you however to actively work on shaking this habit and investing more time and effort in trying to solve problems by yourself before you give up and ask for help. Getting yourself unstuck when you get stuck is a skill in and of itself, and involves important sub-skills like learning how to identify when you have a serious misconception about a problem, learning to believe in your ability to solve problems by yourself, learning to be attentive to small details, and probably other things that are equally important but that I would have a hard time articulating in words. When you look up the answer or ask for help on SE, you end up solving a specific short-term problem (figuring out the answer to the specific question you need solved), but deprive yourself of broader opportunities to acquire these very valuable problem-solving skills. It’s certainly not too late to start though, and your tendency to over-rely on help sounds fairly mild in any case, if it even exists. Good luck!

I wouldn't worry too much. It sounds like you are making excellent progress. You are still an undergraduate, you have tons of time ahead of you!

There are lots of good ways to learn mathematics. Talking with others (including over the Internet) is one. Allowing yourself to get stuck, and trying hard to come up with your own proofs is another.

If you feel that your study habits have skewed too much towards the former, I'd recommend trying out the latter approach. (Which does not mean you have to change to it permanently.) For example, choose a course or book and try to get through it without SE. See what happens!