What equipment for live stream on youtube, facebook, vimeo?

Question

YouTube

Good afternoon,
I run a language blog and an online language course. There was an idea to add live streams to the paysite.
There is light - softboxes.
There is sound - Rode videomic pro.
But the problem is with the picture. There is a panasonic G7k, iPhone 7. From iron - mbp13 2015, asus i7 4/8 c 640 2gb ram and 12gb frame. Adapters (of which a wagon and two carts are on the market) require a percentage of min. i5 4 generations for 4 cores.
What can be bought for the price / quality to transfer pictures from the hdmi port of the camera?
What cameras can both broadcast via hdmi and record at the same time? The maximum that I can pull for money is a6500.
What comfortable upload should I have for 1080p 30?
What pitfalls can be encountered?
The solutions that I found cost 500-800 euros for the "adapter" camera-computer-internet itself.
Is it possible to connect the camera directly to the PC graphics card? There are 2500k with rx470 and 20 GB frame - and in this way to avoid unnecessary costs until there are funds for normal equipment?
Thank you in advance for your help.