What is the benefit of opening port 8333 to a client? I am running a bitcoin core (v0.15.01) full node according to https://bitcoin.org/en/full-node. They say 


  "To support the Bitcoin network, you also need to allow incoming
  connections."


I hesitate opening ports because having a port open for incoming connection allows for anyone to connect. So is this really necessary for bitcoin to work? They to not explicitly state what kind of support this enables.

I see that the client already makes connections to various domains like

x9.dnsseed.bluematt.me
x9.seed.bitcoinstats.com
dnsseed.bitcoin.dashjr.org
x9.seed.bitcoin.sipa.be
x9.seed.btc.petertodd.org
x9.seed.bitcoin.jonasschnelli.ch


Do these represent the "distributed" properties of Bitcoin? Why are there not more of them? Or is this completely unrelated?

Note: I have read questions like this one or this one, but they more focus on the how to, than what the benefit actually is.
If you don't want to, you don't have to open ports at all. You'll still have a full node, it will just have fewer connections. Full nodes usually make 8 outgoing connections and can have many more incoming if you have the port open (for a total of 125 by default). Incoming connections can also be SPV (phone) wallets, or other full nodes.

So the benefit is that you're helping out the network a little bit more providing connection opportunities and bandwidth to more other full and non-full nodes. There's not really a shortage at the moment though.

Typically only the first time when you run the software it might do a DNS lookup on the domains you mention. Each name represents dozens of IP addresses of random other full nodes (not just nodes run by those people). Once your node has a connection to a few peers, it will learn new IP addresses from those, maintaining its own list of IP addresses which is stored when you close the software shuts down.