Hey, my dd was very similar to this. When she had a tantrum we'd pick her up (peel her off the floor), and we didn't lift her to our shoulder height, she would be lifted to waist height horizontally to us, so she wouldn't whack us in the face. We too used time out. Lots of walking away from her but within sight of us, as there was no reasoning with her. We just let her explode.
When she'd refused to follow us, we didn't chase her, we went on our way with her within sight, she'd soon whinge and catch us up. If we were somewhere really busy, and she was playing up, not listening she'd get put in the buggy/shopping trolley.
We had to take dd for hearing tests. All was fine, but she had her adenoids removed for sleep problems, but we stooped fighting over her sleep issues, as we tried everything, and it was making us all very stressed out. In the end we just put her in her room to play, and she'd put herself to sleep.
Dds speech was slow on coming, she was almost mute. After taking her for tests (in Oz), nothing showed up, apart from she shoved a pea up her nose and gad to have it surgically removed 
I did buy a book by Dr.Christopher Green about toddler tantrums, and took some points from there, as I'm not a believer in following everything a book tells me.
Do try getting help from your gp, as a just in case. Good luck 