बूढ़ा means 'old' (as in the elderly). So, you can use बूढ़ा for comparisons only among aged people and even then it may be rather rude because you would seem to be insinuating how 'old' one looks/feels in addition to the numerical age.
जवान means young/youthful. So, when you use it in comparisons, you are not only saying that someone is younger but also that they are more youthful.
When you use बड़ा and छोटा, you are strictly talking about the numbers. You can say उम्र में बड़ा/छोटा if you want to specify that you are talking about the ages and not size/status though it is understood even without that in the absence of any other context that would suggest otherwise.
You can say बड़ी in that case too because it is referring to Julia and not the car. But since you are comparing a person with a thing, there is ambiguity about whether you are speaking of ages or sizes. This would probably be a good use-case for उम्र में बड़ी.
However, for 'My car is older than Julia', you can't use बड़ी. If you use पुरानी, it would be rude to Julia. The only option is probably to reword the whole sentence. Either मेरी गाड़ी की उम्र जूलिया से ज़्यादा है - 'The age of my car is more than Julia's' or मेरी गाड़ी जूलिया के जन्म से भी पुरानी है -'My car is older than (the time of) Julia's birth'.