Does it hurt my chances of getting admitted to check in with colleges on my status, whether they got my documents, when they’ll be notifying, etc?
If your colleges provide a portal that allows you to see that your application is complete, and it IS, then you should NOT check in to ask if your documents have arrived. One main purpose of such portals is so that admission folks don’t get thousands of annoying phone calls or emails asking if an essay, recommendation, test scores, etc. have gone astray. But if there’s no such portal (and you aren’t notified of a completed application in another way, such as via email) then it’s actually important that you DO follow up to make sure all of your application components showed up safely.
It can often take admission offices a couple weeks to update portal information, and it can be scary to see that your material seemingly didn’t arrive when you know for sure it was submitted. So don’t freak out right away if your portal is indicating that materials are missing. Instead, wait about two weeks after all materials have been sent before there’s any need to follow up … or to worry.
As for asking about the notification date … most colleges publish this on their websites or include it with information sent to applicants. But if you really have no clue when you’ll hear (and if you’ve made an effort to find out online), it’s okay to contact the college to ask.