On your asterisk box should spit out the networking information including subnet (assuming DHCP is in play or you set it manually). Ideally you should be setting up a fixed IP and DNS, but you can use the IP in your SIP client.
I think you're opening yourself open for potential pain if you intend allowing inbound connections via your home router to your PBX. VoIP hacking is rife, so if you don't lock it down correctly and you get hacked, you better be on prepaid.
It's also worth pointing out that I have seen cases with simple NAT routers and even some more complex firewalls with fancy adaptive SIP NAT features where they will allow unsolicited inbound connections. In the case of the router it doesn't validate that the inbound connection had a state, so just assumes it's part of the same existing outbound connection and allows anyone to connect in on port 5060. In the firewall case the SIP NAT bypassed all the usual firewalling "to make things easier" which effectively resulted in the same problem as a basic router.