These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
His behavior throughout the series is that of someone completely detached from the real world—with Ted even having to draw him a chart describing the differences between fantasy and reality—and moreover Dougal's thoughts and words do occasionally conform to reality and to varying degrees. In addition, many of his behaviors throughout the show are not outright stupid but just plain weird and nonsensical. He probably only became mentally ill sometime in his adult life, after becoming priest.
Broken Base: More like "broken periphery demographic." Catholic priests either think it is hilarious and harmless satire or insulting and poorly researched slander.
Ending Fatigue: The Christmas special. Even the show's creators complain about it in the DVD commentary.
Ensemble Darkhorse: Bishop Brennan, who gained legendary status after appearing in a total of three episodes.
Fridge Logic: How on God's green earth did Dougal become a priest? Frequently asked In-Universe, especially by Ted. Of course, given some of the other priests out there in universe, Dougal is hardly the worst of them.
The flashbacks of Father Jack in his days teaching in a Parochial School during the first series episode 'Grant Unto Him Eternal Rest' definitely qualify as a "Funny Aneurysm" Moment given the subsequent scandals in Ireland regarding exactly this sort of abuse in Irish Catholic Schools and Orphanages.
One of the Bishops dying of a massive heart attack as lead Actor Dermot Morgan would later die from the same thing.
Even worse, Ted grabbing his chest in shock when Mrs. Doyle surprises him with a cup of tea when he turns on the light.
And the three 'Death' Tarot cards he chooses from the fortune teller's deck in the first episode.
Averted in the finale; the writers originally intended the series to end with Ted commiting suicide, but this was replaced with a clip montage when Dermot Morgan died.
Another cringeworthy one is Jack's P.E teaching.
Teaching Nun: He's also asked me to remind you that it is very hot today, so there will be no need for your tracksuit tops.
Mexicans Love Speedy Gonzales: As noted above, it's incredibly popular in Ireland. Yes, it was created by Irish writers, had an Irish cast, and was set in Ireland, but it was still intended for British audiences, making its popularity in Ireland an unintentional (or least secondarily intentional) byproduct.