Interesting thought, it makes you think that because why does the chest keep track of total weight ? I'm dubious there is a weight mechanic but its either not added in game yet or it's very forgiving at the moment.

Given you can reduce the weight of a cart by jumping on and filling your pockets with cargo, I'ma say probably not. :D

Strictly the number of connects away from a "blue" piece as far as I am aware. For regular wood it's something like 9 or 10 connects to red i think.

so far the only connection for str.integrity I have notice is a separate gradient for the stone objects and wood (blue, green, orange, red... and everyone's fav. dark-red) ,
aside from that - I don't think there is anything else that have any impacts on structures (such as: amount of players/monsters on given block, chests weight....hp of object or w/e)

No. You're good. There's no effect on it.