The plural is 'furniture'
Is "furnitures" a real word, or just "furniture"? I would probably say "tables and chairs are items of furniture" or "types of furniture" or even just "are furniture".
Furniture is non-countable. Types or items are countable, as you suggest.
It is not a real word.
"Furnitures" doesn't exist
Just a note, I've never seen 'furnitur' in Indonesia. 'Mebel' or 'Perabot/Perabotan' seem way more common.
You cannot say 'furnitures' in English
Why is ada incorrect?
Because "ada" doesn't mean "is/are", it means "there is"