1) Does Blackmagic Design Studio Camera 4K Plus only work on 4k settings? Would it work with my atemi mini pro? If so, what settings would I need to change?
The Blackmagic Design Studio Camera 4K Plus is specifically designed to work with all the ATEM Minis through HDMI. The HDMI output can either do 3840 x 2160 (2160p) or 1920 x 1080 (1080p) and you can choose that setting in the Monitor tab of the camera menu. The ATEM Minis are HD only so you will have to choose 1080p for the camera output for it to work with the switcher.
Now, the Studio Camera 4K Plus and Studio Camera 4K Pro will both only record in 4K UHD resolutions (3840 x 2160) in the Blackmagic RAW codec. I believe this is the only option to keep the cost down by not having to pay licensing fees for Apple ProRes.
2) When streaming in 4K do you need to have a specific upload speed? I currently have 15mbs.
If you're using an ATEM Mini then you will only be able to stream in 1080p HD. The ATEM Minis will not do 4K formats at all.
If you are using another encoder and switcher that allows for 4K streaming, then you will need more upload bandwidth. The minimum bitrate I recommend for 4K formats in 24p or 30p frame rates is 15 Mbps, and you need at least 20% more upload bandwidth to handle that reliably.
Lastly, what lens would you recommend with the Blackmagic Design Studio Camera 4K Plus?
You can see my lens recommendations at the link below:
The Best Cameras For Live Streaming - Micro Four Thirds Lenses