Vets can be rubbish at behavioural advice (not all but yours is!) . The water pistol may work short term but you will have a whole load of issues for using it. Dog running off, dog hiding and still eating stones where you can't see him or squirt him and as said before good luck with your recall it will be non existent.
There are so many things you need to be working on which will help on his general behaviour and level of concentration . Please please do not use the water pistol unless you want problems at a later date. By later date I mean tomorrow not in a few years.
He is not listening to you because he is bored and finding his own things to do eg eat stones and leaves.
I say this all the time but it is so important and will make all your training so much easier for you.
You need to get your puppy concentrating on you. Simple to do just keep the puppy on a lead indoors in the quiet and when he looks at you eye contact then you reward with a treat. I do say good when he looks at me but no command. It needs to be his choice. No luring, no pulling him into position. Do this every where in the house, when watching tv, when you are having coffee etc. I would be using at least two of his meals a day on this activity for a few days or week.
Getting technical we are working with classical conditioning which is incredible powerful. Your dog will learn that, being near you is the best thing ever and if I look at you it is even better.
On a practical level if you are out in the garden and your puppy is picking things up and sees you he will come to you for a treat. He is less likely to be picking things up as he is looking at you for a treat.
Teaching heel work he will be looking at you so will not be able to pull ahead.
Recall will be a piece of cake as he is looking at you for a treat and knows that being near you is the best place to be. On another note for recall if you do put a bowl of food down for him always says "Come" in the voice you will use for recall. Do this for the weeks he is not allowed out for walks and you will be halfway there for recall.
Puppies do need a lot of supervision and I would be limiting his garden use unless you are there with him to stimulate him.
He has learnt to jump out of the puppy pen because he is bored and not stimulated. So if you do use a pen again put him in the pen with either a kong or a licky mat to occupy him. If he can not go in a pen then he needs supervision on a long line in the garden - he is not mature enough yet to make the correct decisions out in the garden Eating stones can kill puppies.
Take him into the garden with a long line on a harness and play tuggy with him - that will tire him out and then he can relax and sleep. He should be sleeping at least 18 hours as a puppy more if a young pup.
90% of dog training is controlling the environment so that the dog gives you the correct behaviour rather than correcting the wrong behaviour.