Real people navigate complex situations like mathematical research using metaphorical or visual thinking and honed intuition developed through experience. A lot of it is unconscious reasoning that we form ad hoc rationalizations for upon demand - part of mathematical training is being able to polish such mindstuff into abstract, logical arguments. Part of what makes proofs beautiful, when they are beautiful, is that they 'get to the heart of things' and illustrate why things are the way they are. (Proofs aren't always conceptual and illustrative - they can be also by dry, tedious and unenlightening, and sometimes they can be elegant instead because they cleverly sidestep any requirement to see what's going on.)
Computers won't have the same kind of creative process. However, surely we could be able to make them guided by some kinds of non-arbitrary processes that makes their search more fruitful than random brute search through spaces of strings of symbols (which would presumably be inefficient to the point of being a worthless endeavor). Additionally, a computer's type of creative searching might be able to find things that humans weren't able to find before it.
I think it's unlikely computers will be able to creatively solve the problems mathematicians believe are the greatest, like the millenium problems or Fermat's last theorem or the classification of finite simple groups, even if they can aid us in rote calculations like for the four color theorem. However this is just idle futuristic speculation about how close we can (and will) get computers to simulate human or similar thinking (qualitatively), and I don't think there are any definitive facts at hand.
Computer calculations can do more than rote checks (proof by exhaustion), though. They can do symbolic validation, for example on solutions to the quantum three-body problem of the hydrogen molecule-ion, or find serendipitous numerical patterns that can later lead by human effort to new results, like the BPP formula for digits of $\pi$ or the Lorenz attractor or Feigenbaum constant. See the Wikipedia article on experimental mathematics for more information.
The last question seems to fall under the purview of cognitive psychology of mathematics. I think most of the work in this area has been done on more innate thinking present in children and geared towards analyzing educational outcomes and strategies. However some surely is aimed at higher mathematics and what brain structures are responsible for what mathematicians do and how the relevant mental processes can be modeled.