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Just two words start with LL, and both came into English from Spanish.
LLAMA and LLANO. That is the page. A takes the third box in both, and the fourth and fifth boxes are where they part company.
LLAMA is the animal and everybody knows it. LLANO is a treeless plain and rather fewer people would reach for it.
Both repeat the L at the front, so every word on the page carries a doubled letter. That means a visible double tells you nothing you did not already know.
M and N take the fourth box, one each, and A and O close the words, again one each.
So a single confirmed letter anywhere past the third box gives you the answer immediately. There is no guessing strategy to describe.
Whether your game accepts either depends on the dictionary it runs on. Both are ordinary entries in the larger word lists.
Yes. English borrowed both from Spanish, where LL is a normal opening, and built none of its own.
Not in everyday speech. It means a flat grassy plain and turns up mostly in writing about geography.
As a rule it does not. These two arrived complete from another language and kept their spelling.