Theoretically grammatical, but that wouldn't mean that you eat using a plastic fork - it would mean that there's a plastic fork with you, eating as well.
Jem obiad widelcem - I'm eating lunch with (using) a fork.
Jem obiad z żoną - I'm eating lunch with my wife.
Polish doesn't use any preposition before X when you mean that you're doing something using X to do it.
That would be a participle, "Jem używając plastikowego widelca" - it would sound rather strange, although correct.
Not really. That would mean that the fork is sitting next to you and it's eating as well.