He sounds lovely and so do you. :)
I'm disappointed that the rescue should release him to a new home unneutered and without a current vaccination document though. Very poor standards on their part and tbh the type of rescue which gets others which do a bad name... as you say, many folk have odd misconceptions about rescue and rescued dogs and this sort of practice does nothing to allay such concerns. :( It makes me wonder if the establishment is a pound rather than a rescue although sadly some rescues still operate like this too. Most don't though, it must be said and going by the fact that he was handed in by his owners rather than a stray chances are this is one is a rescue which just doesn't follow best practice (although pounds do sometimes take "surrenders" as well as strays).
Just in case this establishment is one which takes in the council's strays please, if heaven forbid you ever can't keep him don't return him there as his life will be at risk. Instead seek a rescue which is no-kill, which vaccinates, neuters and homechecks potential owners. In fact even if they are a rescue, given that they aren't the most responsible, if that situation were ever to arise, please seek a more thorough one which will do all these things, for his sake.
Time, patience, love - you're clearly giving him all those and I'm sure he will repay you a hundredfold. Please remember to supervise him with children at all times - I'm not suggesting that you won't or that as he's a rescue he will be a problem, far from it, I'm an independent network rescuer and also a volunteer for a rescue so don't think like that, but he's unassessed with children, came with no history and caution should always be taken even if these things have been done and I know how easy it is to get carried away in the excitement. :)
Now , I must tell you that I'm very disappointed in you...
... you haven't told us what type of pooch he is, what he looks like and you haven't offered us drooling, jealous doglovers a photo on your profile! :o