I work in gambling and we write everything in c++. It's as fast as anything else, and the new standard makes everything a lot easier. Threads especially. You also have tons of libraries at your fingertips: GSL, good RNGs, whatever you can think of. I think C is probably the worst thing to write in if you're just coming back to it. You'll end up spending more time with memory management than you will actually getting stuff done. Go with C++ and use standard containers.

I think a lot of companies have this problem, where they buy software the way one would buy a train. The train will last you 30 years easy, so you can spend 2-3 years with specs. No piece of software will last you 30 years, so you can't expect to spend money on it only every 30 years and take 3 years to spec it out. You need to make incremental changes, otherwise you get these massive migration nightmares. My company is going through a simple time management system upgrade and it's a complete mess. People are just not used to change, and they've done it the same way for 15 years. So now it's a giant project, all so people can tell the company when they were at work.

Physically counting paper ballots in front of all the parties' representatives is nice, but it just doesn't work for all systems. I live in Canada, and as it's been mentioned in other posts, our ballots are very simple. The cost of the count is reasonably low. I don't think you can really do this with a Washington State ballot for example. One that goes on for several pages and includes a couple dozen separate votes on a multitude of issues. Your count may take days or weeks and cost a lot more money. If you're willing to spend that money, great, but at some point the cost might outweigh the benefits of accuracy.

It really depends on the application. A class can only teach you the math needed to solve problems many people have encountered already. I do some programming for a gaming (gambling) lab, and sometimes you just run into weird stuff. I don't think you need to be a mathematician, but it's nice to be able to tell what kind of math you'll need to solve a given problem. Then you can usually look it up. Unfortunately, this only comes with breadth. The more math classes you take, the more likely you'll be able to recognize a good approach. I run into problems all the time where I don't know how to solve something, but at the back of my mind I remember some prof going on and on about something that at the time seemed useless but now all of a sudden seems critical.