1. Does being connected increase the chance of religion auto-spread?
I am pretty certain that Religion will NOT (automatically) spread at all unless there is a Trade Network connection (different from a Trade Route) between the City receiving the Religion and said Religion's Holy City.
Trade Network connection = the icon with three arrows in the shape of a triangle
In other words, if the Buddhist Holy City is blocked off from us getting a Trade Network connection with it, we won't be able to get Buddhism via auto-spread.
For example, if the Buddhist Holy City is surrounded by Peaks without a River (or possibly World-Built Roads) running through said Peaks, then no Buddhism can auto-spread to any player on the map.
Err... maybe ZPV's terminology of "plot groups" is slightly more accurate than Trade Network connections, since there is such a concept as a one-way plot group connection, but probably in most cases the two are synonymous with each other.
Note that the requirement for knowing Sailing for a Roadless Trade Network connection to occur is waived if the River connection is within your Cultural Borders.
2. Are you suggesting AH before hunting?
You only really "save" Flasks if you plan to research Hunting before you can receive Hunting in trade.
You might also make more in Food (which often cascades into getting more Hammers or Commerce) if you beeline Animal Husbandry and are able to use the tech after learning it.
Having a Horse Resource a bit earlier (if there is one) could help versus Barbs... I'd rather have 1 Chariot than 2 Warriors as long as each of the Warriors won't be doing a full spawn-busting effort anyway.
If, of course, the Warriors were really doing a 5-square diameter spawn-busting (due to no nearby Cultural Borders or other units to reduce the amount of spawn-busting coverage), I'd probably take the Warrior, but if one additional Warrior will only contribute a few squares' worth of spawn-busting, then having the Chariot in place of 2 Warriors is likely going to work out better for us.
We'd need The Wheel to get Chariots, though, even if we were to settle on top of a Horse Resource, since Chariots require The Wheel as a pre-requisite tech.
If we had a Hunting-based Happiness Resource like Fur or Ivory, I'd be more tempted to go after Hunting, but I don't recall seeing one. If we wanted to Archer Choke (which probably doesn't make too much sense with an AI's big fat cross being unreachable for pillaging purposes), then Hunting could also make sense. Lacking those two things, though, then I'd be okay with missing out on the bonus Flasks to Animal Husbandry in favour of getting Animal Husbandry's benefits a bit sooner.
EDIT: Oops, I forgot about the Deer Resource. Okay, if we need Hunting for the Deer, we can get it next, but if we're okay to wait on improving the Deer, then Animal Husbandry is probably better than going for The Wheel next, in my opinion.