That smacks of a graphics problem, especially as you mention there is a
screensharing issue as well.
Do you have an alternate video cable you can try - either HDMI or
Thunderbolt (is that on the 2018 mini, I forget?)
Also, is there another monitor you can try, especially a lower res one
where the demands won't be so intensive? If so plug that in temporarily
and see if you can reproduce the Time Machine issue.
As for the disk format, I thought modern Macs were happy with older
devices formatted HFS+, just that you couldn't format new ones and that
includes TM drives. At the very least I would expect it to put up a
warning message to say not supported.
--
Bruce Horrocks
Surrey, England