Cherry melomel questions - when to add fruit / vanilla?

Hello Mathematics and Welcome to GotMead!

First off I'd say sweet cherries are not usually recommended for fermenting. They tend to leave a medicinal "tastes like Robitussin" flavor. Tart cherries are usually the way to go. Montmorency is a highly regarded cherry for this purpose, if you can find them. You can still wind up with a bit of the cough syrup effect, but it should age out faster with the tart cherries than the sweet cherries.

The amount of cherry is not out of line. Some folks would use even more. ;D Up to you really, more cherry in the must should equate to more cherry flavor in the end. Adding the fruit later in fermentation and fermenting cooler both tend to preserve the "fruity" character.

Same idea for vanilla. 4 beans is a little higher than I'd go, but the flavor will depend on when you add them, how long you leave them in, temperature, etc etc. I would personally add maybe 1-2 beans either at the end of primary or at racking. Taste weekly, more often if the flavor is getting close to what you want. I'd say probably 2-4 weeks of exposure will give you most of what the beans can give you. Rack or remove the beans when you like it. If it's not vanilla enough for you after some aging (say 6-10 months), you can always add more vanilla.

4-5 weeks is more than you need for the contact with the fruit solids. Usually a week is plenty. You'll pull all the good stuff out of the fruit solids in the first 3-7 days, after that they won't contribute anything useful and can actually provide pockets for spoilage organisms to hide. Best to rack off the fruit after a week or so.

When using fruit be mindful of the cap. You can search for "cap management" for lots of info here. Basically: mix that sucker up well, and often.

I'd not add the acid blend until the end of fermentation, and only if it needs it for flavor. In fact I'd hold off until close to bottling time. The acid balance can be adjusted at any time to provide the "bright lively" character; adding it earlier serves no purpose and can actually hinder fermentation if the pH drops too low. You may find you don't need any acid at all.

When working with fruit, a wide-mouthed container is your best friend. Less chance of explosion, easier to get in there and stir it up, easier to clean at the end. Buckets are great for this. Your fruit contact time will probably only be a week or so, so don't worry about the oxidation menace that people like to mention regarding buckets. If you go the carboy route, give yourself lots and lots of headspace and keep a close eye on that cap. I don't think I know anyone who has used a carboy with [a significant amount of] fruit, and then not used a bucket next time.

As far as timing, I like using fruit in the primary or very early secondary (mead still fermenting, basically). You will probably get a refermentation if you add the fruit to secondary. Lots of folks split the fruit between primary and secondary, some put it all in secondary. Even more people switch methods based on what they want it to taste like. As I said earlier, later additions will preserve more of the fresh fruity character. Earlier additions will lose some of that flavor and take on a more vinous note. Adding some early and some late gives you the best of both worlds.

If you can't find fresh cherries, dried cherries also work. Trader Joe's did carry dry Montmorency, not sure if they still do.

Where are you located?

Dried cherries will give a different flavor, but it will still be cherry. Oskaar has some good rehydration of dried fruit tips if you wanted to look up some of his recipes.