Hi there,
My 16 month old (same birthday month as yours :) is just starting to express pretty specific likes/dislikes/when full/when hungry food interests. The fact that he eats less with you I might actually take as a sign of trust, - that with you he feels comfortable enough in your relationship/communication that he could tell you when he _is_ hungry and you would feed him. Perhaps with grandma and/or dad he is not so sure so figures that he better eat when the food is offered?
Regarding the temper tantrums, we are starting to experience some of those. So far, they mostly come when he is already bothered by something (but doesn't know it) like being over tired, under-the-weather, hungry or, teething.
I always try to make a point if he is starting to scream to pay extra attention because maybe there really is something he is trying to tell me - and if the time demands it - I get down to his level and talk to him there. When telling him there are some things I don't want him to get into we have used:
"That is not for the mouth, it is for the hands(or feet)"
"Not to touch"
"That is for big people, you are still little"
"What are you doing?"
and if he is throwing something that is not to be thrown we say
"That is not a ball! Why are you throwing that? Where is the ball? Go find a ball."
Initially we started by saying these things and taking his hands away from whatever he was doing that we didn't want him to do and giving him something else to play with (redirection)
These days, if he is opening a drawer or cupboard that is not okay with us we will tell him and show him what IS okay (ie. he has his own drawer in the kitchen that I put things okay for him to explore, take out, bang on etc.)
I only use NO! for really important things. I have noticed when I do use it - it REALLY affects him. He gets pretty upset and demonstrates that in a lot of different ways. I know that children get told no in so many places, I really try to reserve it for the most important things (electrical outlets, opening hot oven etc.) and even then remind him at other times that it is not to touch in general.
Mainly, I have found that explaining WHY I am telling him something seems to satisfy much better than a flat no. I think there is a danger in oversimplifying our communication with little ones sometimes that leads to their fustration on multiple levels?
Good luck!