thanks to tell us your knowledge and your goal, this is usefull to know which sort a explanation i need to do(deeper or not).
So, you're welcome, we are here to share knowledge...

EDIT: First check your /etc/resolv.conf, this file should have at least this

nameserver 8.8.8.8

1: lo <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisk noqueue state UNKNOWN mode DEFAULT group qlen 1000 link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
2: enp0s31f6: <BROADCAST, MULTICAST> mtu 1500 qdisc noop state DOWN mode DEFAULT group default qlen 1000 link/ether 4c:cc:6a:b9:98:c3 brd ff:ff:ff:ff:ff:ff

First, good news your internet card is recognized.
Now , i need to know different things

Do you have configured correctly your /etc/s6/s6.conf? show me the file please.
Do you have a correct symlink about localtime in /etc/localtime? Show me the output of the command

# ls -la /etc | grep localtime

Do you have configured correctly your /etc/resolv.conf? show me the file please.
Do you have a correct /etc/hosts file? show me the file please.

NOTE: to see a particular message on the dmesg output, just use pipe like

$ dmesg | grep "error"

platform regulatory.0: Direct firmware öoad for regulatory.db failed with error -2
cfg80211: failed to load regulatory.db

this is concern wireless stuff, not the trouble for the moment.

So, try something, instead of using the service , try to launch dhcpcd directly. first disable dhcpcd

# s6opts disable dhcpcd

now launch it directly on the command line

# dhcpcd -Bd enp0s31f6

this command ask to launch dhcpcd without running it in the background(-B option) with more debugging message (-d) on the interface enp0s31f6. you know the name of your interface when you do the command "ip link show" like you did before.
Now let dhcpcd do the stuff then go to another console (ALT+CTRL+F2 to go to the tty2, tty1 is taken by dhcpcd, we don't not have access on it because dhcpcd was launched in foreground), then open your /etc/resolv.conf wiht nano

# nano /etc/resolv.conf

and append the file with

nameserver 8.8.8.8

exit from nano and try to do a ping

# ping www.google.com

the output should be something like

ping www.google.com
PING www.google.com (172.217.25.36) 56(84) bytes of data.
64 bytes from syd15s02-in-f4.1e100.net (172.217.25.36): icmp_seq=1 ttl=54 time=35.4 ms
64 bytes from syd15s02-in-f4.1e100.net (172.217.25.36): icmp_seq=2 ttl=54 time=35.4 ms

tell me what's happen.
to stop your dhcpcd daemon, simply return to the tty1 (ALT+CTRL+F1) and type CTRL+C.

Writing this tuto makes me think that your /etc/resolv.conf file is not well configured.

Name : yaourt
Version : 1.9-2
Description : A pacman wrapper with extended features and AUR support
Architecture : any
URL : https://github.com/archlinuxfr/yaourt
Licenses : GPL
Groups : None
Provides : None
Depends On : diffutils pacman>=5.0 package-query>=1.8 gettext
Optional Deps : rsync: retrieve PKGBUILD from official repositories [installed]
Required By : yaourt-gui

sorry i'm very busy and i will not be free before monday. So, i answer you quickly.
copy the directory /etc/s6-serv/available/classic/dhcpcd to /etc/s6-serv/available/classic/dhcpcd-mine then change the run file as it :

# !/usr/bin/bash
exec 2>&1
exec /usr/bin/dhcpcd -B enp0s31f6

change the file log/run as it

# !/usr/local/bin/execlineb -P
s6-setuidgid s6log
exec -c
s6-log n3 t s1000000  /var/log/s6serv/dhcpcd-mine

create the directory /var/log/s6serv/dhcpcd-mine and give it a good permissions

# mkdir -p -m 0755 /var/log/s6serv/dhcpcd-mine
# chown -R s6log:log /var/log/s6serv/dhcpcd-mine

then start your new daemon

# s6opts enable dhcpcd-mine

this is should works
note : verify your resolv.conf

So i have a little time to answer you.

First, dhcpcd-s6serv and dhcpcd-s6rcserv was updated on the repo, those services should works now for everybody.

i guess i could put usedwap = on? as i did make a swap partition.

If you have made a swap partition at your installation time and you have a good entry on your fstab then, yes, you can set this variable to yes.

Your localtime and hosts looks good.

warning: /etc/locale.gen installed as /etc/locale.gen.pacnew
warning /etc/s6-serv/enabled/rc/compiled/current installed as /etc/s6-serv/enabled/rc/compiled/current.pacnew

this is absolutly normal, refer to this page for futher informations https://wiki.archlinux.org/index.php/Pacman/Pacnew_and_Pacsave

ah and maybe found a bug, when I come to the Obarun startwindow (bootscreen)

not really a bug, this is know : https://wiki.archlinux.org/index.php/Syslinux# Reboot_and_power_off

hmm I restart the computer. no connection. Have activate dhcpcd manually. Checking resolv.conf: "nameserver 8.8.8.8 " is deleted! file is empty again...what the f...
after putting it in again, everything works, strange

hum, test with the new version of dhcpcd service and tell me if the trouble is persistent.

Is it a bad idea to install yaourt on obarun? Should I select all of them?

Always choose package coming from Obarun for a same package between Arch repo and Obarun repo. If you have obarun as first repo in your pacman.conf, the first choice always correspond to the Obarun repo, so by default the choice is good :).
For sure you can use yaourt on Obarun, the simple way to build and install it

$ pacopts aur -I yaourt

about dhcpcd : to be clear between us. have you disabled your dhcpcd-mine and tried the new dhcpcd-s6serv package?so :

# s6opts disable dhcpcd-mine
# s6opts enable dhcpcd

your issue with your resolv.conf is really strange. Have you another network manager working on your system like Networkmanager or connman?

About your su password : if you use su command like this :

# su

this command switch the user to root and ask for the password, what's append ? the password works?

I'm suprised about pacopts because normaly the scripts detect if a package is already builded and ask you if you want to overwrite it.Well, pacopts is really simple and yaourt have a lot of more features, in this case it's up to you. Personnaly pacopts do the job for me but i know that some user prefer yaourt.