Do you need to be good at computing for a career in physics?

Agreed, knowledge of computers was useful even 30 years ago, and it's close to indispensable these days. In addition to simulations and control, there's a massive amount of computation involved in data analysis, and high-energy particle experiments require massive computation (including hardware-assisted) just to throw out the insane quantities of "noise" events (uninteresting ones). The current generation of telescopes already generates more data than the scientists who are directly involved can analyze; "citizen science" has discovered new things (including a couple of exoplanets, if I recall correctly) by analyzing publicly released data after the fact. The next generation of experiments will produce at least an order of magnitude more data.

But don't stress about it too much right now; start learning, but know that you'll be able to pick up a lot of what you'll need in classes and internships in college, too. You can start with scripting languages like Python, or more traditional languages like C/C++ or Java or JavaScript, or even specialized languages like SQL (used to slice and dice online data sets) or Mathematica/Matlab; it's all good background. Be aware, however, that scientists tend to trail the leading edge of computer science by quite a bit, so don't expect to see much in the way of functional languages, and don't be surprised if you do see some, er, dinosaurs (FORTRAN, maybe even BASIC in some places). If you find yourself interested in low-level details, learning at least one assembly language can be a good foundation for understanding compilers, optimization/performance, debugging, and possibly even writing low-level drivers for specialized scientific equipment.

Yes. It's all simulations and computer-controlled experiments.