They’re usually used with a theme like an Alice in Wonderland theme or a baking theme, usually with Sweet Lolita. The trick to not looking like you’re wearing a costume is basically the same as with animal ears, make it make sense. You can do this by either wearing it with a theme as I’ve mentioned or you can make it a more stylized apron like the heart-shaped aprons you often see so it looks like an actual part of the outfit rather than just something you threw on to do chores. (They can also work with more historical child-like looks, sort of going back to what I said with the Alice in Wonderland theme. Just be careful not to go too crazy with the theme or you’ll look like an Alice face character rather than Lolita.)
I actually think it looks a lot better with the apron to be honest. Aprons can work from time to time in Sweet Lolita and this is one of those times. I really think it looks too plain with out it and if you were to wear it without the apron I would suggest adding a belt or something to make it look less flat.
It depends entirely on the rest of the coord. They can look very cute with some Sweet Lolita coords but it can depend. The first question you should always ask yourself about it is “Does it make sense with this coord?” If there’s a baking theme in the coord or something like that, an apron could fit in quite nicely, the same would go for an Alice in Wonderland themed coord (just be careful to keep it Lolita and don’t just let it become an Alice costume). But if there’s no obvious reason to have an apron in the coord, it usually isn’t going to make sense and will therefore throw everything off.
The first one to come to mind is the one on this dress (it’s detachable.) Most of the ones I’ve seen are brand so they’re a bit more pricy. (I assume by affordable you mean you’re on a lower budget.) I’ve also seen patterns for them though so you could make your own if you sew.
Either way depending on the coordinate. In Lolita, aprons tend to actually work the best on printed dresses because on solid colored dress it’s easier for people to confuse you for Alice or a maid. In the case of this JSK I think the apron is cute but it isn’t going to work for everything. If you’re looking to tone the dress down, take off the apron, but if you want a super sweet, possibly OTT look, the apron can be a cute addition.
It can work okay if you use the right apron with the right dress. Look for things that don’t have many frills, maybe just a simple design. Also be careful with the accessories so it doesn’t come off looking like a maid. Another thing to try instead of a regular apron would be a dress that is fit to look like an apron. Here are a few examples of things you could try: