In Arabic repeating a word is simply a confirmation or an emphasized confirmation that intents to clear any doubt about the importance of what is said or to make clear that this is meant and nothing else.
Here a definition of التّوكيد (at-Tawakyd: confirmation) in Arabic grammar (translation by google translate):
هو أسلوب لُغويّ يؤتى به للتّحقيق وإزالة ...
Answering your question whether the Preserved Tablet reaches beyond the Last Day.
You already cite the most important source
He said: Write what was decreed about everything till the Last Hour comes.
The Preserved Tablet ends with the Last Hour. The decree ends with the Last Hour:
Sunan Abi Dawud Hadith 4708 (EN:4691)
One of you will do the deeds of those ...