When Sam is tapped because he quested and then engages a enemy who has a higher threat than him, he readies and gains +1 on his stats. Now if a second enemy should also engage him, do his stats go to +2 even though he doesn't ready (because he always was ready) or does it stay at +1 no matter how many times a qualifying enemy engages him?

From what I understand, his stats will stack regardless of readying.

You only get one optional engagement per round, but if you can use a card effect (Son of Arnor, The Hammerstroke) to engage more enemies whose cost is higher than your threat, Sam will benefit from them all!