It's about 1/7,000,000 of global emissions. Less than any country I believe (except maybe Vatican City, but that's not listed on this map). Larger than many small towns and airports.

Has to depend on how you allocate. If you count Catholic Church funding of overseas missionary trips and allocate those all to Vatican City, it probably ends up pretty high or maybe not "high" but at least not the lowest of all countries. If you allocate them to the country of origin of the donations, then probably not.

Sadly most of the datasets I'm seeing lump the Holy See in with Italy.

An interesting breakdown would be "carbon emissions controlled" - the US "controls" way more emissions than the US directly makes, as so many emissions from other countries are direct results of US spend.