I have a standard self-propelled mower with bag. Recently I've just been mowing with the "trap door" in the back shut and no bag.

Will this damage my lawn? Does it matter for this type of grass in particular?

Wedge the flap open on your mower. There's no specific mowing rules on zoysia other than keep the height to between 1" to 2" http://www.pikenursery.com/stories_php_loc_02_03_06.html

i heard recently that leaving the cut grass particals help send neutrients back into the lawn and helps fertilize it. You can also use the grass clippings on other gardened areas.

First thing to do is be sure to have mulching blade on the mower. Cuttings left behind can become the fertilizer for your lawn as they deteriorate, they also become the mulch for water conservation on your lawn. No need to bag the cuttings.