I would say that " lleva azúcar?" is the closest to "contiene azúcar?"
When asking "Tiene azúcar?" there is some ambiguity
Tiene usted azúcar? Quisiera poner una poca más en mi café
Tiene el café azúcar? Voy a poner un poco si no tiene ya.
You could have the same ambiguity with "trae" and even "lleva", but due to the usage of the language it is stronger in "lleva" than in the other options. "Lleva" or "trae" are perfect options, although "contiene" would be the best. They are all pretty interchangeable, and maybe the only connotation to distinguish between them is what you are talking about (a recipe, a plate, a precook meal...).
Asking about a plate or recipe I would use "lleva" or "tiene". For something manufactured (somethings that comes from a can for example) I would use "tiene" or "contiene".
Also, can "con" be used? If yes, should with be estar or ser?
Not really, unless you were asking
Se hace con queso
Viene con queso / patatas fritas?
like asking "Does it come with ...?". In the first one you ask if a certain ingredient is used in the process. The second one is "does it come with..."
Note: It may be also interesting to check the answer in the duplicate question "Is that with..."? which also addresses some of the point of your question. Quoting that answer: