I think you need the CLI to do what you want. But here is a quick example.
Step 1: Configure your DHCP server to update the local hostfile.
set service dhcp-server hostfile-update enable
Step 2: Configure your DHCP server to hand out the EdgeRouter as a DNS server.
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 dns-server 192.168.10.1
Step 3: Pick a local domain-name. Set it as the system domain name and hand it out as the client domain-name in DHCP.
set system domain-name internal-net
Step 4: Configure a local DNS forwarder (it will make use of the host file)
set service dns forwarding cache-size 300
set service dns forwarding listen-on eth1
set service dns forwarding name-server 208.67.222.222
set service dns forwarding name-server 208.67.220.220
Step 5: Create a static host mapping in both DHCP and the system hostfile (optional if you want a device to get the same IP every time - useful for port-forwarding).
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 static-mapping ap1 ip-address 192.168.10.10
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 static-mapping ap1 mac-address '00:0f:66:c:79:dd'
set system static-host-mapping host-name ap1 inet 192.168.10.10
Full example:
set service dhcp-server disabled false
set service dhcp-server hostfile-update enable
set service dhcp-server shared-network-name ETH1 authoritative enable
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 default-router 192.168.10.1
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 dns-server 192.168.10.1
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 domain-name internal-net
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 lease 86400
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 start 192.168.10.100 stop 192.168.10.254
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 static-mapping ap1 ip-address 192.168.10.10
set service dhcp-server shared-network-name ETH1 subnet 192.168.10.0/24 static-mapping ap1 mac-address '00:0f:66:c:79:dd'
set service dns forwarding cache-size 300
set service dns forwarding listen-on eth1
set service dns forwarding name-server 208.67.222.222
set service dns forwarding name-server 208.67.220.220
set system domain-name internal-net
set system static-host-mapping host-name ap1 inet 192.168.10.10