Looks like you are American, so here's the deal.
You need a bachelor's degree to get the work visa.  Period.  (Unless of course
you have 10 or more years of teaching experience somehow, then you don't need
the degree, but with that much experience, no eikaiwa will hire you, and you
are not qualified/degreed enough for other places.)

 -what- exactly  are they looking for
when they are hiring a teacher? 
What type of teaching are you talking about?  Universities want people with
master's or PhDs plus publications.  High schools want people with degrees and
experience.  Eikaiwas generally just want a warm body.  Even you degree major
doesn't matter with many.

Does it increase my chances somewhat that I am fluent in Japanese?
No.  Japanese is not used in the classroom anyway.  It will help you in daily
life, tho.

What kind of visa is required to go there?
For teaching English, a work visa, with a Specialist in Humanities
classification.  Or, if you go with the JET programme, a work visa with
Instructor classification.

 I want to live there for a couple years, maybe less. Is there a minimum stay
for visas?
Visas are granted for 1 or 3 years.  They may be renewed endlessly, but whether
you GET renewed is up to immigration.

Would it be possible for me to just move there and find private teaching jobs
on my own?
No.  You need a work visa to work (or a spouse visa, dependent visa, or student
visa).  Otherwise you are working illegally.  Besides, just picking up privates
isn't that easy.  You can sign on to some of the online registrations places,
but you won't be able to live off your privates.  Too unstable.

Is it possible to live a decent life on an income of (approx) US$40,000?

Teaching jobs run 250,000 yen/month (or 3 million yen/year).  Do the math. 
That's about US$24000.  You have to define "decent".  With this salary, you can
expect to have about 70,000 yen left at the end of every month if you go out a
couple times a week and have no outstanding debts at home (student loans, car
payment, etc.).

Is it recommended or required to open a seperate bank account there, or would
my American Bank suffice?

You could open a Citibank account, but your employer would usually open an
account at a Japanese bank for you.

Also would I be able to use my computer, and misc. electronic equip. over
there?
Go here for info on voltage requirements in Japan.  Voltage doesn't change, but
current does, depending on location.  http://www.japan-guide.com/e/e2225.html
Basically, laptops have built-in power converters.

 I know I need a special adapter but are the adapters reliable and is the
voltage the same as U.S. standards?
What special adapter?  See above.  Even the number of prongs is the same.

I would appreciate any info, tips, advice... from Americans currently
doing this in Japan.
Send me an email if you have more questions.
ghill18299@aol.com