No, I neither downloaded any file except the DOGE-Coin wallet nor opened anything in skype. It seems very strange to me too ...
Strange indeed. Legit OS copy or a pirated version? Those might have trojans, backdoors as well. You dont need to answer this, just consider it an option. Other possibilites I can think of are:
#1 infected machine in your LAN that used an undocumented exploit on your the unpatched OS (infected after/durring setup).
#2 infected infrastructure (e.g. your router) - this would also need some sort of vulnerability of your OS or local file sharing
#3 data leak (e.g. shared the data via dropbox with your colleges)
#4 someone had hardware access to your machine (made the TX from there/copied the wallet.dat/installed something)
#5 drive by infection
#6 you use(d) WinXP or older