When you are at the web, where you see the yellow/orange button?
If it is at the top of the web, yes you should see it, if not try to attach a screen to see if anyone can help.
Try to look for :
YOUR PLAN IS CURRENTLY: (ACTIVE)
If you see that there is no problem with your account on the web
And for the engine, I see in your last log you said Launcher ver: 1.0.0-2039063 but you log said:
LogInit: Version: 4.0.1-2031728+++depot+UE4-LauncherReleases
So try to uninstall the launcher, download the lastest from your account and cross fingers.
Good luck!
Edit:I see that you run it with the QFE2 option too, so try to dont use that in the new installation of the launcher