Can the Elapsed callback of a System.Timers.Timer be async?

Will the timer await the callback before resetting the timer?

No. There's nothing it could await, because the signature of ElapsedEventHandler has a void return type.

In other words, your code is equivalent to:

var timer = new System.Timers.Timer { ... };
timer.Elapsed += Foo;
timer.Start();

...
private async void Foo()
{
    ...
}

Whether that's acceptable for you or not will depend on your context. In general, having async void methods or anonymous functions makes them harder to test and reuse - but the ability was precisely given for the sake of event handlers... You should consider how errors will be propagated though.

The title of the question is specifically about Timers, but if we look at it as "How to call an async method after some time?" then you could do it without using a timer.

var task2 = Task.Run(async () => {
    while (true)
    {
        try
        {
            await MyMethod2();
        } catch
        {
            //super easy error handling
        }
        await Task.Delay(TimeSpan.FromSeconds(5));
    }
});

...

public async Task MyMethod2()
{
    //async work here
}

Please note however that this will have different timing (timer will be called at an interval, the code above will be called every (run time + sleep_time), but even if MyMethod2 takes a long time it it won't be called twice. Having said that, you can calculate how long to await for to run 'every x minutes'.