Well, Fertility Rites isn't THAT powerful. Since it doesn't generate faith, it can't guarantee a religion unless it's backed by Piety or a wide faith empire, which is 1.) hard to do by the time you'd get Fertility Rites and 2.) Suboptimal, since growth is less valuable in a wide empire. Goddess of Protection is somewhat highly placed for its worth, I'll admit, but I would take it (in most situations) over Fertility Rites or something of that value. And, for local yield values, I'd say that Stone Circles and Tears of the Gods are a little highly placed. I've seen the AI go after Stone Circles when it had 2 quarry resources but was in desert. I've also seen the AI take Tears of the Gods over Sacred Path when it was in Jungle with 2 Gems.
Fertility Rites is poor because it applies to Growth, not base Food. If you're generating 20 food in a 9 population city, Fertility Rites gives you +0.2 food (10% of the 2 excess food you're generating), not +2 food (would be 10% of the total food you're generating). Sun God is almost always a better pick.
Goddess of Protection is usually suboptimal for its purpose compared to a hammer pantheon (God-King, Craftsmen, God of the Sea) because those extra hammers usually represent an extra 10-20% hammer production earlygame when you'd be using Goddess of Protection for defense. The extra hammer production lets you produce defensive units and buildings faster, both of which are a lot more useful for defending against an attack than a 30% strength boost to cities (equivalent to about 15% damage reduction).
The occasions when you saw the AI picking up Tears and Sacred Path were more likely due to plot owner multipliers: after calculating the added value the AI would get on a tile from a pantheon, the AI multiplies this value by a certain amount depending on whether the tile is unowned (3), owned but not being worked (5), or owned and worked (8). It also multiplies the value by 2 if it would come from an improvement, for some reason. As a result, it doesn't matter if the AI could get more faith out of Desert Folklore than Stone Circles, if it is already working those stone tiles, Stone Circles' faith boost will be multiplied by 16, while Desert Folklore tiles' faith is multiplied by 5 or 8 depending on whether they are being worked or not. Since Stone Circles grants to 2 faith from each quarry while Desert Circles grants 1 faith, this essentially means that one worked stone tile is worth 4 worked desert tiles for pantheons, which is probably why the AI will choose Stone Circles over Desert Folklore. As for Sacred Path vs. Tears of the Goddess, it's most likely because the AI has so many unowned jungle tiles around its cities that they outweigh the two extra faith from Gems: the AI treats culture and faith to be equally valuable when it comes to Pantheon selection.
I'm sure attacking a city with GoP with archers will fail most of the time. I think rule #1 of an early rush is to avoid civs with GoP, and/or Oligarchy(and cities on hills). The quickest way to win, or loose is via domination so I would say GoP being high up the AI's list isn't a bad call. Unless it is isolated of course.
Oligarchy only applies to outgoing city damage, which already reduces city strength by 60%, so it's actually only a 20% damage boost to city attacks (0.4*1.5 = 0.6), not the 50% boost the policy claims. Goddess of Protection's 30% strength boost does not actually make a difference during earlygame rushes: you won't be using archers, you'll be using Chariot Archers (10 ranged strength for 56 hammers, compared to archers' 7 strength for 40 hammers or CompBows' 11 strength for 75 hammers). Because of the way combat damage is calculated in Civ5, a 30% strength boost convert to 14.36% to 16.77% less damage taken (depending on how the city's strength compares to the attackers, lower end if city has less strength than attacker), which is roughly equivalent to taking 6 chariot archers' worth of damage from 7 chariot archers, or about one more turn of the city surviving. Chances that if you have 7 chariot archers beating down on your city's door, that one extra turn of survival isn't going to make a difference. Even if your aim is to defend against early rushes, taking a hammer pantheon is more useful because it lets you produce defending units (and walls) faster.
