Your cousin is mostly right, there is nothing to worry about. The Western Cape is the safest region of the country.
Over the past months Cape Town has had problems with the warring "taxi" drivers which are not taxi drivers in the traditional sense that you are accustomed to. These are mini-vans used mostly for transportation of workers from the townships to and from the city. You are unlikely as a tourist to set foot in a taxi or bakkie as such, so even if those guys are shooting each other you're unlikely to be in the vicinity. There was a strike several months back by the security guards, which can create problems but I believe that's been addressed now.
Casual visitors are fine. There is a danger if you have some affluence and you stay for longer periods of time as you can be watched and profiled as a target, and then robbed. Cell phone and camera grabs do happen, so follow your standard protocol for travel in emerging markets and don't be flashy.
You can definitely go outside at night. You are unlikely to be in a very bad area if your friend lives in Cape Town. In secure areas you're pretty well covered by the security detail. For example the waterfront, any major hotel or restaurant and you'll be fine.
Crime expo is a site to expose the extent of crime in South Africa as an attempt to get the ANC government to step up and start to address the proliferation of crime.
A friend of mine and his fiancee were nearly killed in a vacation area of Kwazulu Natal, which in another area entirely, during a robbery. There is senseless violence in the country and leadership is required to reverse the trend.
It's one of the most beautiful countries in the world and Cape Town is great. A few bad apples make it sound worse than it is, the people are actually incredible there.
Your cousin will know Cape Town inside and out and be able to steer your clear of any problems.