Before answering much, I want to point out something very important that this question sounds like it should touch on: the fact that numeric computation is kind of meaningless without a symbolic part.
You don't just dump your problem onto a numeric solver and let it chug unless it's a very, very simple one (even then, someone spent a lot of time at the chalkboard designing that part of your CPU that adds two numbers). Normally, you have to pose your problem in as clear terms as possible, often using a pencil, and let it ferment. Can you simplify it? What's important? Anything that will hurt accuracy? Is it even posed right? If this is for work, is it worth your time? Has someone done it? Can you do better?
Then you pick a method to solve it. Maybe your first go you'll use a canned algorithm. Maybe it leaves something to be desired, and you start to specialize. Maybe your problem works much better with something slightly awkward, like a constraint, and then you have to extend your canned solution.
Often, almost all of the hard work is done by a guy with a pencil, paper, and keyboard; the labor of the computer actually making it real is worth peanuts in comparison. If not, then either the computation is very successful, the work was botched, or it was something that there's just really not a lot of space to do much symbolic work on (like machine learning).
You might think that as computers progress this becomes less true, but I disagree. As computers get more powerful we throw harder problems at them, these harder problems require more manual effort to even try to approach. That manual effort is a mix of programming and, yes, symbolic manipulation.
- I don't think anyone doubts the outlook of symbolic computation, except maybe folks hunting for exact solutions to nonlinear problems. I wouldn't respect much a person trying to solve a pendulum's motion symbolically nowadays, unless it was for school.
- No, symbolic computation can't replace numeric. Think about it, something as simple as making a picture brighter is honestly a numeric computation. Even if it consists of just multiplying millions of numbers by a constant, no human can do it.
- Yes. Be careful though, just because you have an exact solution doesn't mean it's better; sometimes they're too long or incur inaccuracy (just the other day I replaced an exact integral with a numeric one and got better accuracy).
- Can't really answer this one. Too different, and yet too intertwined.
- Yes, I'd say anything worthwhile. Read almost any paper on anything computational.