I found out that my problem with spinning dots on checkout page was caused by plugin Under Construction. Plugin was loaded when there was no admin logged in, so this was the reason that checkout page worked fine when I was logged in and it didn't work when I was not logged in.
Mompracem, thank you for you suggestion which made me go disabling every plugin step by step and I hope, you will find the reason for this unwanted checkout spinning situation, soon too.
Http://avilera.com works for guests now too.