How do I use DAP plastic wood filler safely?
The flash point of a material is the temperature above which it will give off vapors that can ignite.  It's NOT the temperature at which the fumes or the material will self-ignite. As a point of reference, you likely handle gasoline regularly and its flash point is -45F!
Just ensure that there are no ignition sources while you are using the product and have plenty of ventilation (should not be an issue at an outside door) while using it to prevent fumes from building up.
Answers to your questions:

The can should have storage information but I suspect it says &quot;cool dry place&quot;.

Again the container should have disposal information but you may want to handle it as you would paint or a similar substance and dispose of it at a hazardous household waste disposal site in your community.

It's NEVER a good idea to wash solvents into your sink.  Capture solvents you wash things with and dispose of at a suitable waste disposal site.

Yes, that would be reasonable.