Going beyond the not so great font, is it really a good idea to mix serif fonts with sans serif on the same page, specifically having (sub)headings one and main content the other? I've noticed this in a few places lately, but it doesn't seem very clean to me.
It's fine.
It's often encouraged. Font foundries (and other sources) will occasionally publish lists of type pairs that work well together, and more often than not, one will be serif and the other sans serif.
Actually it's encouraged, but it still means you have to take time in selecting the combo. Georgia and Helvetica play nice together
It's a best practice for design. You want your readers to notice the differences between content and headers at a glance. And having different font families for each is the best way to do that.
Makes sense, you want to make it as clear as possible to see that they're different. My thinking was that if you're already differentiating by size and weight, why introduce another variation that will take an extra fraction of a second to process?