An allonge is a sheet of paper attached to the note that carries the endorsement, the signature transferring the note from one holder to the next. It exists because notes only have so much room on them. After a loan has been sold two or three times there's nowhere left to sign, so each new transfer gets added on the allonge instead. It's not decoration. The chain of endorsements is how you show you're the party entitled to be paid.
The assignment of mortgage is a separate document, and it moves the security interest, the lien on the house, over to you. That one typically gets recorded in the county land records, and recording practice, fees, and what has to appear on the document differ by state, so ask the closer what your county requires and get it in writing.
So the two do different jobs. The allonge transfers the debt, the assignment transfers the claim on the collateral. You want both.
On originals: get the paper note with the original wet-ink endorsements shipped to you or to a custodian, and track it. A scan is fine for reading terms during diligence. If you ever have to enforce, some states will make you produce the original or explain in a sworn filing why you can't, and that's a lawyer's problem you don't want to buy.
One thing worth pulling before you commit: current property taxes and the hazard insurance status. A borrower paying you on time while letting the taxes go delinquent is a slow problem, because a tax lien can outrank your mortgage.