To amplify the above comment, as a neuroscientist with a computational background: don't try to go it alone.
There are a few reasons for this:
1) Research in the field is done by groups because the main problem in generating an 'interesting simulation problem' is carefully defining a scope and a target. That's really hard to do, and generally involves careful discussions between people with different knowledge bases and priorities. If you can't give a clear and succinct answer to the question "How, if successful, will this research advance the field?" to somebody like Larry Abbott, you aren't working on a 'real world problem.'
2) The state of the field is generally about 2 years ahead of the published literature. Unless you have collaborators who routinely attend talks and meetings, and know what people in your area(s) of interest are doing, it's very easy to wind up on the wrong track.
3) Modeling is only useful if it leads to experimental predictions that can be tested, and so needs to be part of an ongoing collaborative interaction between people collecting data, people analyzing it, and people modeling it. Without the entire loop in place, it's difficult to make useful contributions. Also related: outside of things like gene arrays, and a few other standardized approaches, most data in the field is collected by bespoke setups, so even understanding how to parse a data set requires interaction with the people who collected it.
So to answer the original questions:
(1) There are so many that it's impossible to specify. Very little computational neuroscience these days requires more than a workstation. You need to get into a collaboration to reduce the scope of the question for it to be answerable.
(2) It's probably easier than you think, but again it requires collaboration with somebody who's in industry or academia (the latter is probably easier). There are several people I know who informally collaborate doing neural modeling or data analysis with established labs. There are plenty of researchers who welcome informal collaboration, as long as it's competent.
(3) It really depends on who you wind up collaborating with, and the type of question. Neuron and Genesis are compartmental modelling simulators, which you'll only use if you wind up working with people on the molecular end of the spectrum (ie. figuring out intracellular processes). Most systems level work is done using Matlab (some Mathematica and Python as well).
(4) Get involved with non-DIYers. Find a lab to collaborate with! Go to SFN next year, and/or ICCNS/ICANNS/CoSyne/etc (see for example: http://www.frontiersin.org/events/Computational_Neuroscience). Go to posters and talk with people. If you see something interesting, ask if they'd be interested in collaboration.. or ask them your question (1). It'll probably take multiple attempts to find the right group, but there are a ton of groups out there.
Finally, I'd just like to emphasize that working on 'real world' problems in neuroscience (computational or not) is a time consuming endeavor. If you don't think you'll be able to devote several hundred hours a year at the least, it'll be hard for you to find tractable problems.