Can polo ralph lauren 'sport' shirts be tucked in?

Yes. To be honest I do not see a reason why they shouldn't be. But it does depend on the polo (my way of saying sport shirt maybe due to the fact that I have lived in Germany for the past 5 years and there everybody just says - polo) and on the trousers/jeans/etc. you are wearing. So in general (IMHO) it is def. not a faux pas even if some people might think of it that way. If there is a specific "sartorial case" that you are wondering about then feel free to post pictures.

I often see them worn tucked, but I always wear mine untucked.

does the shirt fit you. if it does then you can wear it tucked or untucked and it will look good. if not go get it tailored. an untucked shirt just looks less bad

Shirts should always be tucked in regardless of their level of formality. Untucked shirts always look sloppy.

some desire the sloppy look

isnt that why they have the extra long tail?

Twelve models in two photos and not a single sport shirt (or shirt of any kind)...

Mike, any shirt that is physically long enough to tuck in can be made to look good that way. I've come across button-ups that were cropped enough in the body there was no way to keep them tucked in, they'd come all the way out if you so much as bent partway over.

Tucked, always. It could just be me, but it looks a little bit "amjack" otherwise.

Honestly, as framed, your question may be better posed in SW&D. In MC, the question is
whether and when you can wear it untucked.