with that setup I should use a static IP?
Actually whether you use ethernet or not network gear should always be configured with static IP. The reason is that network infrastructure is passing out IP addresses to clients by DHCP but should really keep the hardware bits of the network static.
It is simple to do.. go to the Internet tab and click Connect Using and change it from DHCP to Static.
Is the first airport the main router in your network? Or are both of them bridged behind another router eg ISP supplied modem router?
Whatever you use as the main router will have a set range of IP addresses.
For example the Airport is going to use 10.0.1.1 by default and in router mode will set DHCP server to use addresses from 2-200

You can then set static IP above the DHCP if you wish.. so 10.0.1.201 will never interfere with allocation of addresses and will then never change address. This helps no end when you need a router or access point or say a printer to reliably work on the network.
And can you elaborate on what you mean by "correctly allocating domain?"
Domain is trickier.
Your ISP might pass the main router a domain that will be included when you open DNS page on your local network settings on the computer.
For example I am in Australia and our main ISP here called Telstra passes everyone a domain Telstra.com.au
This will likely be passed by DHCP in the router to the computer.

So you will have the Telstra domain set for your local network.. but this is wrong.. because the Airport is locked to local as its domain.
The v6 utility gives you no idea that this even exists.
I need to show you from the previous version of the airport utility. I configured the Time Capsule with a name of tc5c. What you don't know is that Apple append the domain .local to this name.

If you attempt to change the name..

You can change the hostname.. so it is different to the Airport name.. but the domain is locked. So now your network is already going to be in a mess.
Lets say you have a different brand of router.. ie non-apple and it has name like Netgear without a domain.. or picking up the ISP domain.. so it is now Netgear.Telstra.com.au
But the Airport has a different domain i.e. tc5c.local
Even though they are directly linked and working with each other.. if search is set to Telstra.com.au then the airport is now invisible.. since it is on the wrong domain.
To fix the problem you want the main router to pass the same domain as the airports are on. Since you cannot fix the airports.
With Apple router as the main router this is not a problem.. but with other brands mixed into the network you do have a problem.. so I always set the domain on the main router to local.
eg.. my Gargoyle router normally leaves domain blank. But I have changed it to the correct domain.
This screenshot is the router configuration page.

And in the computer you should leave the domain blank or set to local.
Your DNS should also include the router IP address but may also include ISP dns or public dns like google.
Actually managing this can be difficult as it does depend on the router you use and how it's particular DHCP works.
Sorry this is complicated.. the chief problem is that apple set their routers with a fixed unalterable domain.
One way you might pickup the issue is with ping command in a terminal.
If you ping the name of the main router and it does not respond.. and cannot be found.. ping the name with domain included.. if it then does respond.. you have a domain problem.