In short (I could elaborate on Nikkor compatibility longer, but actually suffer from a broken wrist)
The old AI mount issue was caused by the aperture ring close to the body where the more modern camera bodies had their coupling lever, therefore pressing and damaging it.
When adapting an old manual lens to a mirrorless body (regardless of manufacturer), you do that via a mechanical adaptor, so no problem.
Quality of the setup of course relies to a significant part of the precision the adaptor is manifactured.
Mechanical interference with lenses can stem from attachments, like the tripod hook for longer 70-200mm lenses, that would collide with the FTZ adapter, as I might recall, or similar obstacles. Also the AI "rabbit ears" that allowed coupling to earlier camery bodies could in some cases scratch the lower range bodies of the F-mount DSLR at the posithion of the flash housing.
(Scratches myself a D5000 with that)
Good news here, that you could unscrew those because they are not needed on a digital body except for nostalgic purposes.
Metering: Here it gets tricky: An AI lens (or AI converted by exchanging or modifying the aperture ring) transmits the f/stop position to the body via a mechanical lever.
Due to marketing decisions, the entry-Level F-mount cameras (D3xxx, D5xxx, and older D50/D60 etc) did not receive that. Those bodies which did receive that, were usually able to do measuring in "A" mode, sometimes restricted by firmware to some simple metering modes. (Matrix metering etc. did also require the distance being transmitted from the lens), which only was avaliable ffrom the AF-D lens and later, the D stands for disctance)
I have no hands-on experience on the Z platform, as my Zfc still is at the retailer ;-)
But I adapted lots of glass to my D7100. In a nutshell, metering is a thing of the firmware. It needs to get the f/stop position somehow and its value (could be entered in the F-Mount bodies) and the metering in "A" mode was possible.
Therefore it boild down to two things:
a) does the mechanical adapter you use transmits the f/stop position to the body?
b) Is the firmware willing to take it into account and allow measuring in the "A" mode? (here the manufacturer could have made some dedisiions to omit such functions in the firmware)
From what I know (and did not see on photos), the Nikon FTZ gen1 does not have the AI lever to transmit Aperture positions to the Nikon Z body.
In this case, it boils down to using the old F-mount lens in Manual mode, which is also in most cases the preferred mode for macro shots.