On my AX88U I found that nslookup reports "non-existent" for static v4 ips when there is no manual assignment list and it is disabled. Adding or defaulting a DNS server address has no effect, in either case nslookup reports the right DNS. Once I enabled manual assignment and defined my static devices they showed up with the host name defined in the list.
There is no way for the router to know about static hostnames that aren't asking for DHCP leases. You'd need to manually add A and PTR records for those. One of the reasons that DHCP reservations are better than just assigning statics.
Some devices do not report their hostname when receiving a lease. If you look in the DHCP lease table you'll see * for them usually, or sometimes a name totally different from what you'd expect. Setting the name manually in the client list won't get them into DNS, a manual reservation is the only way. That will create both an A record and a PTR for it.
From my experience, what is shown in the DHCP table is the only DNS record, and if it says * there is no DNS at all. Regardless of what shows in the client list. It gets really confusing, for example, my Blu-Ray Player
DHCP shows *
Client list default is "HonHai" (the manufacturer of the NIC)
I changed client list to "BluRay"
NSlookup, ping, etc will not work to HonHai or BluRay, nor will a lookup of the IP return anything. If I create a static lease for it, both forward and reverse will work to whatever hostname I give it, and that hostname will now show up in the DHCP leases instead of *
Another example, my TV DOES report its hostname, as COM-MID1. I can look up that and its IP no problem. I customized it in the client list to "PanasonicTV". That cannot be looked up (reverse lookup of the IP still returns COM-MID1). If I give it a static lease with PanasonicTV, now I can ping that name and reverse will return that name, and DHCP leases now shows my customized name instead of the one the client reports.
As far as the domain goes, I've found if you leave it blank, it will pass through whatever your ISP assigns (though that was a long time ago, maybe the behavior has changed). In my case I have my own domain name but I don't want to interfere with internet lookups for that domain, so I just use "intra.mydomain.net". But home is a common one and makes sense too.