All the below for BtS, AFAIK most of it should apply to Vanilla as well

(Side note: get BtS!

)
1. Does the 'Search this Thread' function work? It's taking forever, but maybe because this thread is 750+ pages long. Hope I don't bog down the server.
It kinda works, but it's a bit sketchy sometimes, both in speed and recall

750 pages is nothing compared to searching whole forums, which shouldn't take that long either.
2. Re: Overlapping tiles (see SS)
How can I get Pasargadae to use the shared tile I've highlighted? because I can't. I read in some thread that a city owns tiles that are 2 tiles directly NSEW of it. Could that be it?
Go in Pasargadae's city screen and click that tile. There's no hard rules about a tile "belonging" to a
city, just to a Civ.
3. Re: Gifting Mechanics
Where can I find quantitative numbers for some of the game's mechanics (particularly gifting and effect to the AI). I remember in CivIII that there was a trick where you can continue gifting 10g for 10 turns and get a good boost in civ relations. In my current C4 (vanilla) game, I gifted a substantially complete map to the civilizations and I did note some '+1' effects to some of the civs, and just one '-1' to "you traded with our enemies".
When gifting, everything is broken down to a single value calculated for the gifted item, which is then added to the "fairness bank", which controls how much the AIs like you due to gifts. Approximate values:
Gold ~ 1 point per gold
Techs ~ 1.5 points per beaker
Units = 0.2 points per hammer
Map = gold value (varies in regard to how complete your map is compared to theirs, usually 0-200, more in extreme cases)
Cities ~ 1000 - a lot (+ separate city liberation bonus for "liberated" cities)
This fairness bank total is then divided by the turns you have known an AI (x10). So on the turn you meet someone gifting 20g is enough for +4 (the maximum), but if you've known them for 200 turns it takes relatively much to impress them

You might notice gifting units is basically useless.
You got different results for gifting your map around because you have known different Civs for a different amount of turns, and they had different completion on their existing maps.
4. re: The Chop
Why is it that sometimes a forest chop will deliver 250+ shields, and other times only 90+ shields?
Chopping inside a city's BFC will always yield 20 or 30 hammers (60, 90 on Marathon) depending if you have Mathematics or not. This amount gets the bonus production modifiers as normal, so if you are buliding Angkor Wat with Stone (+100%), have a Forge(+25%) and are in Organized Religion with the state religion present (+25%), with Math, a BFC chop will give 75 (225 on Marathon) hammers. Chopping farther away gives less base hammers per chop.