Hi and welcome to you and Max. It does sound a little odd that you have no papers, record of vacination etc with your kitten. I would make sure that the vet gives him a thorough check just in case of any problems. I certainly wouldn't take his previous owners word that he has been vaccinated.
As for keeping him amused, my Siamese girl loves to play fetch with a piece of rabbit fur. She also loves playing with 'da bird' which is bunch of feathers on a rod and line. If Max is doing things you don't want him to a firm no and remove him from the situation may be enough but you could always resort to a water pistol - a quick squirt is usually enough of a deterrent. (I use a plant water spray).
Eating, she is not allowed in the dining room while we eat so that isn't really a problem but I know she would try to join us if she could!
Just a warning - Siamese are incredibly curious and your Max will probably try to get into everything from cupboards to the smallest spaces! I had to rescue Mai Tai before she vanished behind the kitchen units via the back of the sink cupboard the other day! If a door is open she wants to know what's the other side.
Good luck with Max. we'd love to see some pictures of him please?

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