Catching errors from nested async/await functions

If etcFunction() throws, does the error bubble up all the way through the async functions?

Yes. The promise returned by the outermost function will be rejected. There's no need to do try { … } catch(e) { throw e; }, that's just as pointless as it would be in synchronous code.

… bubble up all the way to the top-level Promise?

No. Your topLevel contains multiple mistakes. If you don't return the doThing(data) from the then callback, it will be ignored (not even awaited) and the rejection stays unhandled. You'll have to use

.then(data => { return doThing(data); })
// or
.then(data => doThing(data))
// or just
.then(doThing) // recommended

And in general, your function should look like this:

function toplevel(onsuccess, onerror) {
    makePromise()
    .then(doThing)
    .then(onsuccess, onerror);
}

No unnecessary function expressions, no .then(…).catch(…) antipattern (that could lead to onsuccess and onerror to both be called).