If your Schengen Visa only allows you to stay 13 days (during which you may enter multiple times) and you have used all 13 days then you have 0 days left.
With that you cannot reenter the Schengen Area.
Non-Schengen countries that have this rule, expect you to be able to return to the Schengen Area.
Having exhausted the amount of days allowed, you cannot ...
You can only use your Bulgarian residence permit (with your passport) to enter the UK without a visa if the card is for a family member of an EU citizen (член на семейството на гражданин на ЕС) and you are traveling with or joining that EU citizen.
Note that a blue card is a work permit, so it would not qualify. You will need a visa to visit the UK.
For Turkey - as a Lebanese, you are allowed visa free access upto 90 days within 6 months period for tourism only.
For the other two, it should be possible to enter Bulgaria using Romanian short stay C visa or vice versa. There is no clarity about number of entries required on the visa, so trying this with a "Single" entry visa may be risky.
As states ...
According to Timatic, the system used by airlines, your residence permit means you can visit without a visa:
“The following are exempt from holding a visa:
Passengers with a residence permit issued by Sweden for a maximum stay of 90 days”
It is probably a fare that is only offered through a certain agent, not by the airline itself. Airlines sometimes have special arrangements with agents/resellers. Those get discounts for talking a gzaranteed number of tickets, or for dealing with customers requests or other sich things. Not all fares have to be available directly from the airline.
If you hold an EU member state visa, such as Cyprus, you can visit the aforementioned countries as well as Serbia. However, the criteria is different from country to another (required number of entries, duration of stay, etc.). You can fill and check your info here and get the detailed criteria for each country: https://www.emirates.com/ae/english/before-you-...
As you are not the owner of the car, you will need a letter of authorization from the owner. Please contact the rental company for it and eypect to pay an additional fee for it.
Which border checkpoint to take depends on your starting point and your destination. The major one is Kulata - Promachonas.