I could have written your post. I have a 3 y/o son (and 12m old son, too lol). Anyway, all of a sudden Jacob (3 y/o) has started with very similar actions that you mentioned. I know it's a phase and he's testing. But, I can tell you that we never, I mean never give in. I don't care if he fits in the store, at a restaurant (did it at Paul Bunyon's in Wis Dells a couple of weeks ago). He doesn't get what he wants.
I explain to him that we don't talk that way. At first, if he wants a cookie he will say, "Waaaaaant iiiiiiit." I tell him I can't understand him when he talks that way. At first I would have to have him repeat me "May I have a cookie, please." But after (what seems like) a 1000 times, all I have to say now is, "How do we ask for things in this house?" And he'll go into the "May I..." routine.
For the extreme fits (tears/falling on the floor). I let him do it for a little bit and then I tell him if he doesn't stop he will have to go in time-out for screaming. I will get down to his level or bring him in another room and talk to him. I get him to count to three with me and then take a deep breath and then exhale. I usually have to do it alone a couple of times and then he will join in. (Hey, it's good for me, too and really is relaxing.)
I'm not sure about the sending to his/her room. My husband did that recently with Jacob and I've read that you shouldn't. Something about a negative association with their room, maybe. But, I also agree that if s/he's going to be having a fit it's not fair for the rest of the house to have to suffer. I'm on the fence about that idea, still.
So, you're not alone! I think it's just something that we have to be consistent with and not give in. No matter what. Once they realize that the more stink they make, they get stuff - then it's all over. Lol.
Good luck!
EDITED to add: Yea, I think I like Margo's response a lot. Letting them tantrum thrower decide when they are ready to be part of the household. That gives some control and I think they want that. I think I'm gonna try that one. So, thanks for the posting Margo and thanks for asking the question H.!