If, when you login to Fastmail, you do it by opening your browser eg Internet Explorer, Firefox, Safari, Chrome ... - then you go to
https://www.fastmail.fm/ and login .... then your problems have nothing to do with an 'app' running on an iPhone or iPad.
If you do use an 'app' then the problem might be with the app, and it would be worth trying a browser on the iPhone/iPad instead.
Apart from the email problem, you said: "OR my not being able to check out on its site." If you mean that you're able to attempt to buy stuff on various websites but the final stage of this does not work - then that, which should have nothing to do with email - suggests there's something else (though I don't know what) going on.
Could you describe what does happen in a situation like that? For example most such sites let you click on various things you want to buy then will eventually show you a list of the things you selected, in your "trolley", "cart", or "basket". Are you seeing the list of things? If every page has a note at the top (as Amazon does for example) of how many things are in your basket so far, do you see that increasing as you choose more things to buy?
Does the problem affect all browsers?
You also said: "In general, what's not going out is stuff I try to buy. (Except, of course, Amazon.)"
Neither of those statements make sense to me. First ... "what's not going out is stuff I try to buy." Is this different from the checking-out problem? If it's different, what are you expecting "to go out" when you buy something?
And... if it does work ok with Amazon, why is Amazon ok "of course"? What is special (to you) about Amazon?
Have ALL your emails stopped arriving?
A while ago you experimented with some filter definitions which turned out to be deleting the wrong things. Have you set up any filters since then? Have you changed ANYTHING ELSE in the way that your FM account is configured?