this graka is not oversized (1 slot wide) and does not require an additional power connection. Should not be a problem.

I do not know. The HP EliteDesk 800 G5 SFF has a device height of 10 cm (according to the data sheet). A smaller slot bracket is also supplied, so installation should work. I don't see any problems with power consumption. The only thing that could be a bit tight is air cooling. My card is in a midi tower.

Hello
The dimensions of the Geforce GT 1030 are: Length from mounting plate 17cm
width 7cm
Depth 4cm (total)
Since I don't know the PC HP EliteDesk 800 G5 SFF, I don't know if the card will fit.
In my experience, however, it should not be a problem, at least in terms of power consumption
because every PC has a little reserve in terms of power consumption.

Hello, the thickness is approx. 37mm.