My son is starting similiar behaviors. At this age, they just don't have the words to express their frustration and throwing things (including shoes) is their way of saying, "I'm mad/frustrated/upset/etc."
If you can ignore it, that's typically the best option. No audience, no tantrum (or a reduced version of what it would have been.)
Another thing I read that helps avoid some of the behavior is to limit saying no. Instead of, "juice?" "No." Try answering with something else like, "after dinner, you may have some juice." Or, "we'll have juice tomorrow, but now it's time for bed." You didn't say no, they didn't hear no and it seems to help. Obviously there are times when you have to say no, but by limiting how often, I've found it helps. It also helps to stay calm (as challenging as that can be) and talk in a low, soothing voice.
Then there are times (like for us, this morning) where he was throwing the mother of all tantrums and I have no idea why. We were running a bit late and so I sat on the floor with him and talked soothingly while getting him dressed. He was mad and just wanted to be left alone but it wasn't an option for us. I don't even know if he knew what was wrong so he got extra hugs while he just cried. We got into the car and I sang to him soothingly despite his continued screaming (didn't want to be in the car seat either, I guess.) It always works because he has to stop screaming to hear what I'm singing (which is a made-up song with a bunch of animal noises thrown in that always gets his attention and builds anticipation for which animal I'll sing about next.) Worked like a charm despite my completely off-key voice.
As for your mom -- my mom doesn't live with us but is a fan of giving my son what he wants to avoid the tantrums. It's the grandma in her but I've had to tell her that I'll deal with this and she can speak with him when he's calm.
I would recommend a similar conversation at a calm time when everyone is unemotional and things are relaxed. Let her know that giving him what he wants in the midst of a tantrum undermines your authority as the parent and doesn't teach your son what you'd like and need him to learn. If all else fails, tell her you're the mother. (I've been known to use that with my mom and although very confrontational, cuts to the point and gets the message across.)
Just be patient, hang in there and know that this too shall pass. (Although while we've all heard of the terrible twos...I've recently been told it's really the f'in fours you have to watch out for, so we'll see.) Good luck!