1) One lever moves upshifts and the other downshifts. Take a minute to play around with it and it'll be obvious. Pedal it on a stand and actuate the levers.
2) Wouldn't be surprised that the shifter is broken. Also can be an adjustment issue followed by cable + housing issue
http://www.parktool.com/repair/readhowto.asp?id=75
3) Again, on a bike that's been sitting for a while, you can almost guarantee that the cable + housing suck
http://www.parktool.com/repair/readhowto.asp?id=64
4) All wired bike computers operate on nearly the same principle. Magnet attached to the spoke on a wheel passses by a sensor which is connected by a wire to the mount which is typically on the handlebars/stem. The computer sits there. Battery is secured by a cover usually, again on the rear of the unit. If it is fubar, new wired units with multi-line display starts at around $10-$15.
Your questions are not easy to comprehensively answer without at least a little bit of homework and effort on your part.