I have an allergenic group of friends -
Me: Vegetarian, allergies to mushrooms, mangoes, and limited milk (sometimes dairy-free)
Friend 1: Allergy to latex (plates need rewashed due to dishwashing gloves), cross-reactive to banana, avocado, kiwi, chestnut, mango, cashew, one other tropical fruit (forget which - she had a longer list but cleared most of them) - and, at Disney, precut apples (long story)
Friend 2: Allergic to strawberries, peanuts, tree nuts, eggs in large quantities, fresh fish, shellfish, etc, etc.
We ensure 1TS/day, occasionally 2. We do eat some leftovers, but we can usually eat well enough through QS that with the guarantee of one "solid" meal at TS, we do fine. I have actually learned NOT to utter the word "allergy" at a QS if I can help it at all - I have visited enough to know what I can eat and my needs typically do not need chef oversight. Once you tell the QS order-taker person that you have an allergy, they will call a chef or coordinator (and maybe hand you an ingredient binder) and will not order until the chef has come out. I will say I have noticed that they have loosened up a little bit and will sometimes agree to just give me the binder, especially when I mention, "I just need to know if there are mushrooms in the veggie burger, it's an allergy but not anaphylactic or cross-contamination sensitive." However, if we are in a situation where there are worries where we need to see a chef, it is ALWAYS accommodated. One single top-8 allergy is usually very easy to handle, but when you get into combos, it does get a bit more complex. Just pack your patience and know that if they determine you need something specially prepped, it may take a little while extra.
(I have also gone and eaten almost entirely QS and been okay - but we find we are much happier with some TS to break up the day!)
I also might consider not doing a dining plan, so that you can start out with one TS a day, but if you find you need a little extra "support" you can add some towards the end of the trip. (And, honestly, you'll probably come out ahead in the $$ category anyways.) I would avoid buffets and seek out places with menus - and ensure that the menus have dishes you feel are likely to be safe or easily modified to be safe.
(I can tell you that of your tropical fruits, you will encounter banana frequently, but rarely run into mango and papaya. Do you have issues with meat tenderizer? I've been told it is sometimes somehow derived from papaya/enzymes. If so, that would be the only thing to be aware of there. Mushrooms are usually "called out" on the menu with the exception of occasionally being hidden in some veggie dishes - esp veg burgers - and also mushroom soy sauce at yak & yeti. They do have regular soy sauce available. The rest of your list is, of course, much more common and going to cause you more issues.
Another tip - we have also found that when I travel with both of these friends, providing chefs with an index card or similar that has a list of who is allergic to which is most appreciative... especially if you make a copy for each restaurant so the chef can make notes and take it with him. With my group, for example, it saves the chef from doing a double-take when Friend 1 orders mushroom risotto!)
HTH!