The thing is, that the city is a mana ability, and thus the trigger of the cities damage will go on the stack after the triskelion is put on the stack. A city's damage is not the same as the LED's discard effect, which can only be played on instant speed, not on "mana ability speed", where "mana ability speed" implies being able to use the ability after a spell has been anounced.
The City's damage (technically the triggered ability that will make City deal damage to you) will go on the stack above Trike
only if the player playing Trike anounces and then pays for it.
Ancient Tomb's damage is part of the mana ability. It will not use the stack.
The mana ability of Lion's Eye Diamond follows the normal mana ability resolution procedure (resolves immediately, doesn't use the stack); you just can't play the LED ability unless you could play an instant. Comparing this situation to LED is bad because of the timing restricion on LED's ability. The Oddyseey Eggs are better here.
Bottom line: With a City, if your opponent is smart, you won't have a chance to weld in the Sphere to prevent the Trike from being played. If he plays incorrectly, you can.
With Ancient Tomb, it doesn't matter how your opponent plays; you won't be able to stop the Trike by welding in the Sphere.
City of Brass
Land
Whenever City of Brass becomes tapped, it deals 1 damage to you.

: Add one mana of any color to your mana pool.
Ancient Tomb
Land

: Add

to your mana pool. Ancient Tomb deals 2 damage to you.