I guess Prague would suit someone with a family better. Never spend long periods of time there, but whenever I visit the Czech's capital alone, or with friends/family we leave with happy memories.
how often do you have to get these documents and to whom ?
Best option is to rent a real cheap apartment outside Sofia; in some villages, prices can be as low as €150 per month. After renting the apartment, sign-up for internet and utility bills, and keep paying it in order to keep it active.
I guess that you have to find a way to have real consumption for your utility bills (to show someones lives there) to be on the safe side?
Workarounds for that too; but consider that internet and bills are always flat fees, and with agreements with apartment owner, is possible to have flat fees for utilities as well.
Around $2k for non-resident UBO company (all fee included) + high street banking
Do you think there is a possibility to operate through a US LLC from Bulgaria and pay 10%?
It depends, so many variables. For what you need an US LLC? Where's the managing based? Where's the "center of interests" of the company? And so on.
The idea would be to appoint a Bulgarian company as a member of the LLC. Hire a manager for the LLC in Bulgaria, an office... in short, a bit of substance.
PE in Bulgaria pays 10% CIT.