It's ironic that you've made the very mistake that so many people make when they're trying to be "productive". His name is "Brian"... not "Brain". Although many people will overlook such a thing, many more will not especially on "first contact" and may, in fact, judge you by that.
Being productive doesn't mean cranking out a lot... it means cranking out a lot that's right. 😉 Too many people rush trying to be productive and end up making very costly mistakes even in something as simple as an email. If you make a mistake in code by rushing and it gets promoted to production, it will cost much more to find and fix the mistake than it would if you'd slow down and double check/test your work as you go. Possibly insulting a first contact, especially a possible customer, by mispelling their 5 letter name can cause damage that will never be repaired because first impressions are lasting and you may lose the customer because of it.
It is NEVER worth doing over... do it right the first time even if it means you need to be "less productive". Ironically, you'll be much more productive by slowing down in the long run.
So far as emails and instant messages go, I absolutely agree that it should be contained and constrained. I've had people in the next cube send me emails and IM's with no other distribution instead of just saying "Hey Jeff... can you yada-yada...". It's gotten so it's all very stupid. When a message comes in, it's difficult to tell from the "You have Mail" alert whether it's something critical or something that can wait and, of course, people just don't put whether it's something that needs immediate attention in the subject nor do they correctly mark the email priority. As a result, I have to check every time the bloody bell rings because it could be an automated email from a system in trouble or the big boss needs something for a customer right now.