Xperia for Livestreaming + Recording

[Raal43]

Hi! First time poster here

I do a fair bit of videography, almost exclusively on Sony gear (a pair of A7Sii bodies and an A7iii). I often do full coverage for competitive performing arts shows, and recently have been asked to offer live streaming as part of my typical package. To this end, I was looking at picking up an Xperia 5 III. I wanted to understand the process better, but I've struggled to find good technical documentation beyond the 5-minute videos available on YouTube. A few questions, before I pick up the Xperia: 

• Is there good technical documentation available? I'm not afraid to dive into a handbook, but I can't seem to find anything. 
• Am I able to livestream from either the A7Sii or the A7iii in 1080p while simultaneously recording to the card in 4K? While this is not a dealbreaker, this is easily the strongest reason for me to do streaming through an Xperia vs through a 3rd-party app or computer. 
• Should I connect the external shotgun mic to the camera body or to the phone? 
• Is there any camera-side time limit placed on those streams (flexible for YouTube, Facebook, or Instagram lives, since expected output is 1080) 
• The theater temperature is typically well-regulated, but is there a real risk of the phone overheating for livestreams that reach 4.5 hours (the long end of these shows)

I appreciate the assistance for any of these questions. I'm typically a "research and test first" type, but I'm about to launch into a 5-week travel sprint and my phone died, prompting this opportunity. 

Photo is 14mm at 2.8, ISO 800 A7s ii (7iii was recording video)