Yes they haven’t been taught any manners.

... at the table either. Good list.

This should be called Table Behavior as there are NO manners involved.

These children are definitely too old to behave this way. The reason they do is because they have been allowed to do so.

-farting at the table
No dessert

-not using utensils properly
Teach them and make sure their utensils are the right size for their age (smaller forks for smaller hands)

-using hands
Tell them once, then no dessert.

-touching other peoples food with hands without asking first
Without ASKING? NO, NO, NO. IF someone wants to give them food from their plate, they place it on the child's plate. The child does NOT touch it until then. Place them far enough away that they cannot reach others' plates. If necessary, buy a Youth Chair and use that with a dinner tray.

-making a big mess
No dessert or dinner is over.

-throwing food
Dinner is immediately over and it is now bedtime.

-not saying please or thank you unless asked repeatedly
Stop asking. No dessert. No second helpings. No tv time. Etc...

-leaving table without asking while others are still eating and going thru pantry for cookies and cakes
NO DESSERT. Lock the pantry. Leaving the table without asking? Must be bedtime.

-being extremely fussy about a lot of different types of foods and throws tanturms if they dont get what they want to eat
We eat what we are served. Don't like it? Here's a PBJ or a bowl of cereal. Tantrum? Must be tired so off to bed.

The dining-table is a battle ground- kids learn to use it from an early age to get attention. The solution is really very easy: In the natural world parents eat first- it is basic behaviour. You can't help kids until you've helped yourself.

When I was a child we had a pecking order- dad was served first, then mum, then the kids got what was left. If we were lucky, and everyone had their share, you might get second helpings. If you didn't like the food you didn't eat.

The adults talked to the adults and the kids talked to each other, and we listened. It was called conversation- you learned to ask for things to be passed to you, and when someone asked, you passed them stuff. There was constant interchange of words, ideas, and opinions on the world. The food was rarely even mentioned. Oh how I long to be back around that family meal table!

Table manner is taught and learned behaviour.
Such a poor display is a direct reflection of whomever is raising these kids.

You have to find a common ground. My SO had different table manner standards than I did and his were stricter than mine. His weren't particularly strict or horrible - so I bent to his standards and BS had to learn to live with it and now it's no big deal. Do we still have to call him on it sometimes? Yes. SO's kids were raised with stricter table manners than BS and we still have to call them on manners sometimes , too. Especially using fingers in the serving dish if we're serving roast potatoes. Roast potato brings out the animal in them!!! Sometimes you will have to remind kids about what's ok - like the please and thank you - that is normal and age appropriate, although annoying. 

If these table manners bother you - and I agree I would find them bothersome - then you need to have a discussion about acceptable standards and then you have to enforce them.

Re. food they don't like - I just don't bother with that. Everyone at my family table has weird food dislikes, which I have just decided to cater to. Even my SO has weird food issues and I have a couple. Some people take a hard line on that, as a couple, we don't. I insist on kids trying things. But it can be a very small bite.  

We are slowly getting our 19mo trained but even he does better than that, the 7 yo does slip from time to time but besides the interrupting thing (she does it allllllll the time) she does pretty good at the table, minus the whining when she doesn't like something.

I can't tell the age of the second skid (it's replaced by an emoji in your post).  SO's kids are 5 and 3, and here's what they do or don't do, if it helps you figure out "normal."

- farting at the table  - happens very occasionally, always followed by an excuse me.  They know it's not acceptable.

- not using utensils properly - they're working on this, but for the most part they use them fairly well.  I'm not bothered because certainly the younger one doesn't have highly developed fine motor skills yet.

-using hands - happens with some items

-touching other peoples food with hands without asking first - NO

-making a big mess - yes.  Some food usually end up on table/laps and drinks get spilled sometimes.

-throwing food - Never.  The child would no longer be at the dinner table if that happened, and would not want to be anywhere near SO!!  

-not saying please or thank you unless asked repeatedly - Nope.  Please and thank you are a requirement.

-leaving table without asking while others are still eating and going thru pantry for cookies and cakes - Nope. They don't leave the table and they have to ask before they can get sweets.

-being extremely fussy about a lot of different types of foods and throws tanturms if they dont get what they want to eat - Nope.  They're required to try what's on their plates, and if they don't eat it, they'll go without and there will be no snacks.  Of course, when we have them I try to serve kid-friendly meals that they enjoy, and I do my experimenting, making spicy foods, etc. when they're with their mother.

I think the idea of an etiquette class is fabulous.   It's good to know the "rules" and have them taught and enforced by a third party expert.  Of course, there then needs to be follow-up or it won't stick.