It's not very common, no. It is usually used with 'for' and a noun: 'he's a stickler for punctuality' or 'he was a stickler for etiquette'.

For me, I don't hear this word often in daily casual conversation. I would not say it is very common, but it is also not uncommon. If you use this word, people will still understand it.

It is commonly used in speech. Less common in written materials, and probably would not be used in more formal writing. Common Example: "He is a stickler for detail."

Like many phrases, it's in between. You hear it in US speech. It's not rare. Most native speakers understand it. It's neither formal nor informal. It's not "literary." However, you don't need to add it to your active vocabulary. As John says, it's almost always heard in the phrase "he's a stickler for X." You can just as well say "he insists on X" or "he demands X." "Stickler" suggests someone who is extremely insistent or extremely demanding.