If you control multiple "Lightsworn" monsters, their End Phase effects don’t chain to each other (they each activate, one after another).
If you do not have enough cards to send to the GY remaining in your Deck when resolving the effect of a 'Lightsworn' monster or "Judgment Dragon", all cards remaining in your Deck are sent to the GY. For example, if you control "Jain, Lightsworn Paladin" and its effect activates during your End Phase when you have only 1 card remaining in your Deck, that card will be sent to the GY.
If the effect of a 'Lightsworn' monster to send cards from the top of your Deck to your GY during the End Phase is negated by the effect of "Light and Darkness Dragon", that 'Lightsworn' monster's effect activates again in the same End Phase.
If an effect chained to a 'Lightsworn' monster's (or "Judgment Dragon") effect to send cards from your Deck to the GY during the End Phase that removes that 'Lightsworn' monster from the field, the 'Lightsworn' monster's effect does not resolve.