Personally I try to limit the amount of time my dogs play with other dogs. All I want them to be is sociable with other dogs and know their doggy manners. As for play - predominantly I want them to play with ME.

I certainly wouldn't be encouraging prolonged sessions, especially with a smaller dog who may get hurt. My dog, who is very well socialised and pretty much ignores other dogs in favour of me, is a Collie X, and with small, fast moving dogs she can become too highly aroused for my liking and a bit "chasey" and I tend to stop such sessions before they get going.

IMO, yelling and squirting water at him is likely to excite him even more.

Tug toys are fine, and a great way for dogs to expend energy and learn that YOU provide more fun than other dogs. As long as you teach them that the game ends when YOU say so, they are great fun and a valuable motivator for training/teaching.

Giving them toys to "compete" over is, IMO, a bad idea and asking for trouble.