I am not a USA resident, I live in Europe (not EU zone). What would be my tax obligations to the USA if I had LLC formed through Atlas for a purpose of software SAS business (selling software subscriptions worldwide)? I know I would be obliged to pay income taxes in my country, but I don`t know if and what would I need to pay regarding taxes to the USA.
You could owe tax to the state in which you incorporate, if they tax LLC income earned out of state. (Most states don't, but you will have a "minimum franchise tax" that you must pay regardless of LLC income for the privilege of being registered in their state.)
You generally would owe US federal income tax on the income you earn from your US customers. But, in many cases you could qualify for tax treaty benefits that reduce your US tax liability to zero. You would still need to file a US tax return, but you would not actually pay US taxes. (Your tax return would explain why you don't need to pay.)
You would generally be taxed on the income in your home country. Despite common belief, almost all countries tax the worldwide income of citizens residing in their country. (Territorial taxation is generally reserved for corporations and expatriates.)