Extend is implemented in Inkscapes developement version (ready in 0.49)! Assume you have two paths A and B. Path B is ready, path A should be prolonged linear (linear to its handle, that means in the same direction like the handle) and snap to path B. With the node edit tool (F2) drag the closest end node of path A to path B while pressing Ctrl+Alt at the same time.

Trim is a bit more complicated. Assume you have two paths A and B which intersect each other. You want to cut path A at the intersection of path A and B. Then you have to:
• Duplicate path B (select path B and press Ctrl+d)
  (This copy of path B is now over path A; good because the copy of path B is the knife that cuts path A)
• Select both path A and the copy of path B and press Ctrl+Alt+/ (cut path)
• Path A is now divided in two parts. Now remove one of the parts ⇒ trimmed!

The problem: Not very intuitive. But after you have done it once, you will remember always.

For the trim to be intuitive it would be nice to have these intersection tools: http://wiki.inkscape.org/wiki/index.php/Intersection_Tools