OK, it's been a while.  Other things got in the way of progress.
Just to update, for those who may be searching in the future, the mower is running.  I had been hoping to get this running for my mother (her mower), but just couldn't figure it out.  So, it went to the shop.  It had a new starter, new fuel pump, and I had removed every ground connection and cleaned them.
As it turns out, the shop tells me the carb was all gummed up.  It also needed both blades, a new belt, and they're saying it still needs new drive pulleys due to rust.  I was discussing this with my brother and he tells me about how she left the mower sitting outside, uncovered, for about 7 months this past winter.  Here in Washington, it's known to rain a little, and the past couple years we've had ridiculous (for us) amounts of snow.
So, moral of the story....
If you let your mower sit outside in the weather for half a year, you might have problems.  In this case, it cost my mother $257 at the shop to get her mower going, plus $25 for a fuel pump, and $120 for the starter we put on it (which it needed anyway, but probably wouldn't have if it was stored properly).
Thanks for everybodys help.  I appreciate it.
:thumbsup: