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Actually it's encouraged, but it still means you have to take time in selecting the combo. Georgia and Helvetica play nice together

Here's some combos

http://bonfx.com/19-top-fonts-in-19-top-combinations/

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.

It's fine.

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?