I cannot speak for other continents but from Europe the best airlines are:
- Cathay Pacific via Hong Kong
- Emirates via Dubai
- Garuda Indonesia via Jakarta
- KLM, via Singapore
- Qatar Airways, via Doha
- Singapore Airlines, via Singapore
- Turkish Airlines, via Istanbul
All of the above airlines have both flight to Jakarta and to Denpasar
The term best airlines is not for everyone the same. Speaking about quality I would say: Singapore Airlines and Qatar Airways.
It depends on when you want to spend your holiday in Indonesia.
-for the months July and August the best time is to book between September and November the year before; this is from Europe. I guess from other continents it will be about the same.
-for flights between 10 December and 15 January the best period is around May/June
-for flights after 15 January, February, March, April, May and June the best time to book is 3 to 4 months before
-for flights in September, October, November and up untill 10 December the best time to book is 3 to months before
- Garuda Indonesia
- Qatar Airways
This depends very much on your plans before or after Flores. Coming from abroad and when you go to Flores only or combine Flores with Bali, Lombok, Sulawesi, Sumba or Timor the best airport is Denpasar.
Good direct connections with domestic airports are Kupang (Timor), Tambolaka and Waingapu (Sumba), Praya (Lombok) and Makassar (Sulawesi)
In the months July, August, the Idul Fitri period (in May 2020) and between 15 December and 15 January you best book 4 months ahead.
Outside these periods 2 months in advance shoudl be fine.
Flores has 6 airports; from east to west:
Maumere Labuan Bajo, Kupang (Timor), Denpasar (Bali), Tambolaka and Waingapu (Sumba)
Ende Labuan Bajo, Kupang (Timor)
Bajawa Labuan Bajo, Kupang (Timor)
Ruteng Kupang (Timor)
Labuan Bajo Bajawa, Ende, Maumere, Denpasar (Bali)