'Garlic' is uncountable, so you would always say 'pickled garlic'. We do not generally pluralise this word. The countable forms are 'bulb' (for the whole intact object) and 'clove' (for the segments). As for the other vegetables, it would depend on whether they are whole or not: If you have small carrots, baby eggplants or tiny cucumbers (cornichons) which are whole, they are still individual 'things', so you'd probably use the plural: "I'd like a jar of pickled pickled carrots" or "I'll have some pickled eggplants". However, if the vegetables have been cut up (diced or shredded, for example), they are 'stuff', so you'd probably use the singular: 'pickled carrot' or 'pickled eggplant'. By the way, this would be the case whether they are pickled or not. Once a countable thing (an egg, an apple, a potato) is no longer a single whole object, it becomes uncountable (scrambled egg, stewed apple, mashed potato).

Not usually. It would be uncountable, especially if it's cut up in a jar. "some pickled eggplant" However, if they are not cut up, then they may be countable. "two pickled eggplants"