Say you want to travel by air sometime in the next three months, and you are
totally flexible on the date. Which sites can search for the lowest air
fare across a very broad range of time?

www.itasoftware.com lets you search across a month at a time.

Is there a service that would notify you by email when an extremely
below-market rate becomes available on short notice?

Try yapta.com.

R's,
John