Internet connectivity works only after rebooting - Manjaro/Ubuntu

I suspect three possibilities:

• hardware failure (broken components or just unrealiable conections)
• driver failure (dosn't handle some state correctly)
• software failure (dosn't bring up interface because of some feature/bug)

Things to try (You may need to adjust/change commands used):

ifconfig -a
ifconfig eth0 up
dhclient eth0

You can also try reloading driver module using rmmod / modprobe.

EDIT:

Since previous attempts didn't help, I would suspect something hardware related (unreliable connection, fried component, etc). If reboot helps, problem may be related to power - You may try:

• instead of reboot turn off PC completely (maybe even unplug from utility power), wait a few minutes and turn on it again;
• try resetting machine before OS starts booting (turn on, wait until bootloader is displayed and reset);
• try take-out and insert LAN card (if it is not integrated).

EDIT2:

From comments it is clear the problem is related to "TCS fast calibration failed" message during boot. A few things to try:

• check cat /sys/devices/system/clocksource/clocksource0/available_clocksource - are contents different when internet is working and when it is not?
• try adding kernel boot parameter clocksource=acpi_pm or clocksource=hpet (got idea fro this page)
• check PSU for capacitor plague (MB already checked as per comments)
• try removing LAN card - do You still see TSC error?
• if it is an option - try installing some older Linux distribution to hard disk (not CD/USB boot) and see if it works better with LAN card.
• try replacing PSU or LAN card, as these are components which most likely may be out-of-order.

From my experience, I would bet the source is PSU (had seen enough weird problems on similar hardware).

EDIT4:

Seems it is not PSU... If other options above fail, try capturing:

dmesg > boot.log

And check for differencies when LAN is working, and when it is not. You should see at least TSC error which You decsribed before, but there may be other differencies too:

diff bott_ok.log boot_fail.log

EDIT5:

From the comments it seams problem is related to PCI enumaration during cold-reset and hot-reset, identified using lspci while in GRUB loader (before Linux kernel loads). Things to try:

• boot kernel using options like reset_devices, nobios, conf1, conf2 (just a few which got my attention, You can find many more in Documentation/kernel-parameters.txt, look for BIOS or PCI related options first);
• sice machine loads successfully from USB, pluging in USB device while booting from HDD may trigger correct behaviour.

EDIT6:

Clear BIOS settings (reset, clear using jumper on MB, upgrade/rewrite). Try playing with "Reset configuration data", "PnP OS installed", PCI timings or similar BIOS sertings.