Can I trust large companies like Google not to store failed password attempts?

Nobody can ever tell you to trust or distrust some entity; you, yourself, will need to choose to trust this entity. You already trust Google not to load exploit code in your browser, you already trust Google by actually using their browser (Chrome), you already trust Google with your private conversations (GMail), your embarrassing search terms (yeah, they know what you like).

You see, whenever you have a question like this, simply think "Do I really matter?". The answer is most likely "No". You're nothing, you're a droplet in a sea of information. Every conversation you've ever had with anybody, every memory you have, every SMS you sent, every phone call, every email, every photo, all of them put together are less than 0.000000(a bunch more zeros)001% of the data Google and other big entities have and work with.

So, should you trust that Google will not store your failed password? I don't know. Should you trust that the NSA will not have access to that password? I don't know. But here's something I know for sure: The NSA don't really need your password to access your email, they simply tell Microsoft or Google "This guy... national security threat... send me all of his emails", that's it.

So, worrying about it doesn't matter very much. If you really feel like you should worry, then just change all instances of that password. Otherwise, don't think about it and enjoy your day.