Do I need to brush off the bees from the donor frame? Or just put it in the recipient hive as is?

You need to not move the queen, other than that, your choice. If you are concerned about weakening your strong hive then eggs only is the best bet.

"One last thought: Would I be better off starting a new colony by taking 2-3 frames from my strong hive and letting them raise a queen and then adding those bees to my weak hive? Basically doing a split with my stronger hive and then adding them later to my weak hive?"

I have come to prefer the LaFerney split (not named because he invented it, but because David said it in a way that gave me the Duh! moment.) Move the queen and frames away from the larger hive. It is a better balance of resoruces and gives a brood break as well.

If you split it is better to add the layers to the new rather than the other way around. Home court and size advantage. You do not need to add all at once either.

I do think there is a queen there now, just not cranked up yet. Worst case is drone laying Queen, that would be rare this time of year.