[mothered]

Great, I'll show my drivers license and a scan of the written letter. Thanks for that!
What do you think they will do with it? Verify my signature with the one that signed for the delivery?

It's simply part of company protocol to move forward with your claim.

They'll cross-check a few details and If they cannot conclude whether you received the package, the SE will work In your favor.

Hi, quick update and some advice required. I sent them my ID and the hand-written letter with my signature which is different from the one that is signed. I sent this 5 days ago and still no reply. Should I wait, or open a Paypal dispute?

Depending on the Influx of claims, It may take longer than expected.

It won't hurt to contact them, with a gentle reminder that you're still waiting on your claim.