I had finished my Ph.D. (Atmospheric Physics) and landed my dream job. I looked around and realised that I had little in common with all of my new colleagues and quit science within a year. That's when I first heard the term "Quantitative Analyst". I didn't really understand what the job was, but a bunch of people seemed to want to talk to me and some of them were even willing to put me on a plane to come and see them. I eventually landed in the energy industry in Houston and had a great time over the next few years working with really smart people and making good money.
I'll try and answer some of your questions, but strongly recommend you have a read of this book. It has lots of different stories about how people entered the profession and it demonstrates there are lots of paths other than doing an Masters in Financial Engineering (or whatever they are calling it now-a-days).
* The math geek - you can throw anything at him and he will understand the math and build you a model in R or S-plus or some other obscure statistics analysis tool set that only he knows how to use.
* The hacker - loves finding patterns in data, lives and breathes correlations and tells his friends he is a prop trader. In reality, he sits next to a trader who ignores everything he says except the stuff that confirms the trader's pre-existing biases.
* The coder - takes the models built by the math geek and the hacker and turns them into tools used by different parts of the business. If he takes a wrong turn, ends up working in Risk Management (not that there is anything wrong with that...)
* The structurer - one of the most fun quant jobs. Working with the origination (sales) guys, structuring different packages of derivatives to create a new instrument. I spent most of my time as a quant doing this.
If you don't have a Ph.D., then expect to have a lot of doors shut in your face. It's not impossible to get a gig as a quant without a Ph.D., just harder. This is just the inherent bias in the industry - I got one, so why should I hire anyone who didn't...
This is particularly true if you you don't have any experience. An alternative path is to develop a complementary skill (such as coding) and gradually shift into a front-office role by making yourself useful.
I won't talk in any detail about the hard skills, as they should be obvious. Math, coding and concise writing skills are all pretty darn important.
For most quants, the soft skills are where you will earn your money. Influence is key. Building a great pricing model is one thing, convincing a trader to use it to identify nice spread trades is completely another. If you think out of sample testing and running a shadow book is the way to achieve that then you have a lot to learn.
Attitude is critical. If you can't speak in a way that makes people listen, you are going to struggle. If you can't answer a yes or no question in a binary fashion, you are going to struggle. If you start sentences with the phrase "My hypothesis is..." you are going to struggle. This is a tough world full of people who believe they are smarter than every other person in the room. Equivocation is not the way to earn their respect.
It also helps to have a healthy attitude towards money. A trading floor is a great place to learn that relative wealth is a far more significant driver of your day-to-day happiness than absolute wealth. Learn to ignore the money or risk spending 364 days of every year being miserable and 1 day being relieved or furious.
Best of luck.