My son is 31 months old and if what you described isn't normal, then I'm in trouble, too! I do think it's all normal. It is so shocking, isn't it - this barbaric behavior.
The one thing you said that I don't relate to is the not wanting to go somewhere. My son always wants to go - no matter where it is - so I don't know if your daughter NOT wanting to go anywhere or not is normal. I guess it would depend on the reason she doesn't want to go - maybe it's just she doesn't like her car seat or something simple like that. Try asking her.
You say you have an 8 month old, too. I don't have any other children, so I may be off-base, but maybe she is just seeking the attention. With 2 small kids, I imagine your day is BEYOND busy - and just by nature of his being 8 months, your son is going to require more time and attention. So maybe figure out a way to focus on her, and her alone, for set periods of time. With my friends with kids close in age like this, this seemed to be what worked for them to help the older child from acting out so much.
But as for just tantrums in general - yes - they are violent, long, exhausting, overwhelming, shocking, but completely normal. I am sure there are some circumstances when they would not be considered normal, but from what you described, those sound normal to me (but I'm not a doctor and only have one child, so what do I know???). If she is truly NEVER happy (as opposed to just having grumpy days often, and lots of tantrums), if her entire day is difficult with no "happy" times, then you might consider taking her in just in case. My son is so moody, but he does laugh and play, too. If your daughter doesn't do that, then maybe she's not feeling well - she may not have the words to tell you - it could be a tooth, an earache, the heat in general, etc.
As for eating only meat, yep, normal, too. If she is truly not eating anything else, you might consider a multivitamin or something to make sure she's getting nutrients. Food issues are rarely about the food - they are usually about control.
Good luck!