Why do you think Harry and Hermione leave the Invisibility Cloak on top of the tower?

The why is because this cloak had been passed down in the Potter family since the Peverells, as was revealed in The Deathly Hallows. Dumbledore had borrowed it from James (the attraction of the Hallows), and was now returning to Harry his birthright.

Does Harry get his invisibility cloak back?

He evaded Death using the Cloak until he grew into an old man. He finally gave the Cloak to his son, and greeted Death as an old friend.

Who gave Harry his invisibility cloak back?

headmaster Albus Dumbledore
In the first book of the series, “Harry Potter & The Sorcerer’s Stone,” headmaster Albus Dumbledore gifts Harry an invisibility cloak, which belonged to Harry’s deceased father, James. The gift included a note: “Your father left this in my possession when he died. It is time it was returned to you. Use it well.”

Why didn’t the Dursleys let Harry stay at Privet Drive?

They also couldn’twait since the agreement between Harry and the Dursley’s was that they were separating and Harry had no intention of coming back (not calling Privet Drive his homeanymore); this would immediately void all of the protections:

Does Harry ever wear his invisibility cloak casually?

Harry is the most responsible teenager in the world, at least when it comes to his invisibility cloak. He only wears it when he has a good reason to do so. The closest he comes to wearing it “casually” is when he’s sneaking off to the Mirror of Er… Something went wrong.

What would have happened if Harry and Lily had the trace?

Good question; Harry was to turn seventeen within the hournext few days, so the Trace would have worn off at that time, and they would have been free to Apparate, conjure a birthday cake, cast Tarantallegraon each other and have a merry old time, or any magic they liked. But so would have all of Lily’s protection:

Why didn’t they avoid the Battle of Hogwarts by apparating?

Battle comes at cost and there was always a chance that the original Harry would be killed in the battle. Why didn’t they avoid it (or, at least shorten its length) by apparating to a safe place? One more advantage of apparation was that Death Eaters couldn’t see who Harry was even if Polyjuice Potion wasn’t used.