I used to shoot with a photographer who used at minimum 2 lights outside, 1 key and 1 fill. This is something I plan on jumping into soon, I need to pick up a portable battery pack for my strobes.
First 2 shots with the MP look great, along with the 4th one. You can minimize shadows with light placement, output and diffusion, I would try some different diffusers on your flash/strobe units. What are you using for flash?
I have 2 Alien Bee B800 strobes with these attachments: one 22" beauty dish, one 47" octabox, and one 34 x 44 softbox