How can you do this combo at all? The Reaper King is a legendary creature. Wouldn't the token copy retain the legendary rule and immediately die since there would be multiples of the same legend out?
This combo does work. You are correct in that
Reaper King 
is a legendary creature. You are correct in that copies do copy the card types including "Legendary." The game itself enforces the legend uniqueness rule and the rule itself is not tied to a particular creature or permanent.
It works because a triggered ability triggers whenever anytime its trigger condition is met even during the middle of a resolving spell or ability and a triggered ability exists independently of its source.
Rite of Replication 
begins to resolve.
"Put five tokens that are copies of target creature onto the battlefield."
All the triggers mentioned previously trigger but are not yet put on the stack.
Rite of Replication has finished resolving and is put into its owner's graveyard.
Just before any player would regain priority, State-Based Actions are now checked.
The game sees that there are two or more legendary permanents all with the same name and then moves them all to their respective owners' graveyards. This is not destruction. Regeneration won't help. Being indestructible won't help.
After all the State-Based Actions have resolved and there are no more State-Based Actions to resolve, then the triggered abilities are put onto the stack. The triggered abilties exist independently of their sources so the removal of
Reaper King and the copies don't change what happens.
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