You ordered and ate the food. The waiter served you your food and brought you the check. You've paid it (via cash or a credit card). If cash, the waiter will say "Thank you!" when he or she picks it up. If with a credit card, he or she will say "Thank you!" when they leave the paperwork. Either way, as mmontero pointed out above, you would just say "You're welcome!" in response. A cashier is someone who actually takes the money and puts it into the register. A waiter is the person who took your order and brought you your food. They are not usually the same person, so the order in which you list them is not important. Both ways are acceptable. While not a "rule," per se, if you use "a" before one, you should use it before the other, for balance, especially when writing (as you have done!). Speech is more informal and not doing so is okay. Regardless, the response to "Thank you!" is always "You're welcome!" I hope this helps you and clarifies all your questions!

It's appropriate to reply with "thank you" (emphasis on the "you"). If you say "you're welcome", you sound arrogant... remember, they served you. ;)

I always reply by saying "Thank you," with the emphasis on the word you. Think about Mandarin, where the response to "xie xie" is often "bu yong xie" (literally, no need to thank) or "bu ke qi" (it is nothing - IF I understand them correctly). The same is true of Spanish, where the response to "gracias" is "de nada" (it is nothing). When exchanging politeness, too much is rarely bad. To answer your second question, the article "a" modifies both nouns, so "a cashier or waiter" is grammatically correct, just as "a cashier or a waiter" is also correct.

You should say (You're welcome!) but you can say (Thank you!) as well.

I guess you should smile and then say you're welcome.