However, I feel like just asking something like "What type of research do you do?" would just come off as lazy since most professors already have information on their website.
Yes, don't ask anything you could reasonably learn from their web site, their publications, textbooks in this area, web searches, or Q&A sites. It's incredibly annoying when people write to ask me to put work into explaining something to them, when they show no sign of having put any work into this themselves first. (The worst question of all is "how can I apply to your department?", since that one can be answered in a few minutes by anyone with internet access.)
Also, don't volunteer your services for the professor's projects or propose a collaboration in a cold e-mail. This might be appropriate later, after you have established a serious research connection, but until that point it's awkward. Nobody is going to say yes based on very limited information, and the choice of no vs. maybe puts people in a difficult position: if they say no based on very limited information, it may come across as insulting, while if they say maybe it can come across as overly encouraging.
So that leads me to wonder, what are some productive questions that I could actually ask professors in the context of a cold-email or scheduled phone call?
From my perspective, the only really productive approach is to engage seriously with the professor's research area. If you can propose new ideas or questions and start a conversation with genuine intellectual content, then that's great. It will be a worthwhile discussion for both of you, regardless of whether it helps you get admitted, and you may end up impressing the professor. But this is a high bar, and you shouldn't expect to be able to send substantive messages to many people. If you can't think of anything genuinely interesting to say, then it's better not to say anything at this stage (you can think a little more about this area before e-mailing the professor).
The focus should be on the subject matter. When random strangers write to me, I'm not particularly interested in telling them about myself (to the extent I want to tell the world about myself, it's far more efficient to do so publicly, rather than telling one person at a time). I'm even less interested in hearing an infinite series of random people tell me about themselves. Most of them want to be admitted to grad school in my department but aren't going to be, and it's just not a good use of time to read a lot of e-mails in which applicants tell me more or less the same things they are going to say in their actual applications. On the other hand, I'm always pleased when I get an e-mail that starts a genuinely interesting research discussion.