In all three cases, you have to find a balance between how much you are willing to offend them, and how much you care about your own health. I'll start with the easiest one.
We once hired a cleaning lady from a different country. She took the sponge from our sink and used it to clean the toilet, and put it back. Now she used that same sponge to do the dishes.
In this case, you hire someone to do some work according to your specifications. If you hire them to make your place cleaner, the job description does not include sampling bacterial growth from the toilet and spreading it on your tableware. Anyone doing that in a restaurant would get fired immediately.
So, easy case: someone who is hired to clean should understand cleanliness. This employee doesn't have the skills required to do the job. If they consider it normal to clean the toilets and the kitchenware with the same sponge while you watch, just imagine what they do when you don't watch. Just fire them. No need for subtle interpersonal skills in this case, since you're the one paying!
At the self-serve line at the cafeteria, we had a group from another country visit. One girl took the serving spoon, put it in her mouth to taste it, and then put it back on the dish. This must be something they do all the time in that country and they think it's alright.
In this case, I suppose you care more about the person, so you'll have to be a bit more tactful.
It is unsanitary, but you seemed to have trouble convincing her. I suggest using a different angle and getting offended yourself (fake it if you must). The idea is to convince her that it is rude, which should be a lot easier than convincing her that it is unsanitary.
If (as you imply) she recently arrived in the country, learning the ways of the locals is part of adjusting to a new culture, so gently telling her that something is considered rude here is a normal part of this process. It does not require her to adjust her own views and prejudices. Much easier.
You could mention that sucking on the spoon and putting it back in the dish is rude here. If she wonders why, tell her it's like spitting in the dish... which she kinda did, indirectly.
This is similar to good old Gordon's method, which is based on rephrasing stuff like "You're annoying!" into "I don't like it when you do this." The latter has a much higher chance of success for obvious reasons: there is no accusation, and it leverages her will to be nice to you. It has the drawback of only working on people who actually care about not annoying you though, but this should be the case.
In any case, pick your food from another dish.
At dinner, they served from the main dish using their utensils that came right out of their mouth, and used them to put food on our plates. I know they meant well, but we consider that practice unsanitary.
We don't "consider" that practice unsanitary. It is objectively unsanitary. According to WHO, 67% of world population carries herpes HSV-1. I don't, and I like it that way. The stat is high enough that if you make it a habit to share utensils/glasses at parties, then you will be part of the 67%. It is quite inconvenient. Not to mention EBV, CMV, Hepatitis, Strep, etc... and the common cold.
Also, it is flu season. It isn't germ-phobic or silly to avoid contamination and inconvenince by using simple common sense.
The answer was always "it's our culture" and at times, they got offended and said we were too rigid or "we don't know how to get along with people".
This is where things get uneasy. Their retort is a simple and common shaming tactic which aims to shift the blame on you. You don't know how to get along with people! It is an attempt to make you conform to their own culture. However, this happens at your place: your house, your rules.
How to reply depends on how rude you are willing to be. If you never invite them again after this (as you suggest), think about it: it is actually pretty rude, and in a sneaky way. If their cooking tastes great, and they are fun to be with, it's a bit of a bummer.
The most polite thing to do would be the same as the previous one: tell them it makes you uneasy, and you will have trouble enjoying the food. Since it is the flu season, you can even insist that the news repeats every day that we should be extra careful about contamination! Or you can agree & amplify their shaming tactic and pretend you are indeed a bit phobic of germs and they should respect your quirks.
Also, really, licking the spoon and putting it back in the dish is like spitting in the soup.
An important thing is not to fall into the shaming tactic: when they say you "don't know how to get along with people" just laugh it off. And if they say it's part of their culture, this is just a lame excuse: just pretend you didn't hear it.
You can rub it in by microwaving your plate to sterilize it ;)
Personally though, if they still don't get it after the gentle and diplomatic attempt, I would simply escalate to stage two:
I'm French: we have weaponized cheese.