Hi Richard
I'm doing a PhD in Psychology, but with the aim of being a research/academic psychologist. From what you've written it sounds like you are considering something like Clinical Psychology (an extremely competitive field). The British Psychological Society website is quite good for information about the different types of psychology - have you had a look at it?
Regarding becoming involved in people's problems, I can partially answer that, having spent a lot of years working in the addictions field before returning to uni to start a PhD. In a professional 'helping' career there will be support structures in place - formal supervision, employee counselling and so on, to support you in the work you're doing. However, a lot of it is down to you, your own ways of coping, your ability to keep a healthy routine going so you can leave your work behind at the end of the day, and your ability to seek help when you need it. Some situations are inevitably more stressful than others, and you need to be able to recognise it if you're at risk of becoming over-involved or burned out.
I'm not sure what stage you're at in your education or career, but hope this helps!