I didn't test as thoroughly as @MikeMcQ did, but it did seem to fail once, and then work for a while, yes.
No, I hadn't.
I think the consensus around here tends to be that the only advantage GoDaddy may have is that they can be cheap, if you don't mind working around their systems and attempts to upsell you.
There is a semi-official forum recommendation list listing providers that are known to have integration with Let's Encrypt for getting certificates. But of course, there are other considerations one might want to consider when choosing a hosting company. At this point, having good free integration with getting a certificate (regardless of CA) is the bare minimum one would expect.