Travelling Violation Scenarios

Q1: Should a ball-handler stop and fall to one knee be in violation?  Should a ball-handler stop and fall to one knee be in violation, is not a travel, based on the first question above.

A1: Answers to both of your questions are covered in the rulebook by article 25.2.2:

A player falling, lying or sitting on the floor:

• It is legal when a player falls and slides on the floor while holding the ball or, while lying or sitting on the floor, gains control of the ball.
• It is a violation if the player then rolls or attempts to stand up while holding the ball.

Q2: Does a player bending down to pick up a loose ball but have one knee contact the floor and then rise up cause a travel violation? 

A2: Yes, it is a travelling violation, based on the second bullet point noted above.