Should the sound level in Triode mode be lower than in Tetrode

it all depends on load. Both Mack truck and Honda Civic can cruise at 50mph, but the load is what makes it different.

Yes, if you connect a 100dB sensitivity speaker to the 100 Watts amplifier and a 86dB sensitivity speaker to the 250 Watts amplifier, the 100dB speaker will sounds much louder!

if the gain of both amplifier are identical and connect to a pair of identical speakers, the required output power of both amplifier is less than 100 watts to provide that degree of loudness (volume), the output volume of both speaker should be same.

BTW, a 250 Watts amplifier can’t provide 2.5 times louder volume than a 100 Watts. The difference of both amplifier output is about 4dB.
According to psycho-acousticians that a level 10 dB greater usually means "double the loudness" or "twice as loud". In this case, you need 1000 Watts to get twice as loud!

If I would connect a right channel output of a preamp to a 100 Watts monoblock and its right-channel to a 250 Watts one then the right speaker will be 2.5 times louder than the left one.

@imhififan not quite.
it all depends on load. Both Mack truck and Honda Civic can cruise at 50mph, but the load is what makes it different.

1/2 the power is only a 3 dB difference, not much. The amps input sensitivity may also change affecting the volume level. The real question is which mode do you like the sound of better. If you can't tell a difference between the two just keep it in tetrode mode. 

@imhififan 
+1 Well put.

I can connect left channel output of the preamp to a 100 watts power amp and the right channel to 250 watts one, and expect both the left and right speakers sound with the same volume just delivering a different headroom quality?

Yes, if the gain of both amplifier are identical and the required output power of both amplifier is less than 100 watts to provide that degree of loudness (volume).

Let’s put it this way, the max. output power of an amplifier is the top speed of a car, the volume control is the cruise control of a car.
If you set the cruise control at 50mph on two cars, one car with top speed of 100mph and the other car with top speed of 200mph, both car will run at 50mph as the cruise control setting.
Now, if you set the cruise control at 130mph on both cars, the one with top speed of 100mph will run at 100mph and can’t go any faster, the one with top speed of 200mph will run at 130mph as the cruise control setting without problem.

yes

3 db difference, all else equal

That's interesting. By some reason I have always believed that the power amplifier's wattage directly translates to volume gain. Please correct me if I am wrong, but according to what you say, I can connect left channel output of the preamp to a 100 watts power amp and the right channel to 250 watts one, and expect both the left and right speakers sound with the same volume just delivering a different headroom quality?

The mistake you make is that power does not equal gain.
It just means more headroom.

If I would connect a right channel output of a preamp to a 100 Watts monoblock and its right-channel to a 250 Watts one then the right speaker will be 2.5 times louder than the left one. 

No.

FYI, I had the same results with a pair of VTL MB125 in Tetrode/Triode mode that gave me pause as well. @ghdprentice summed it up well. Enjoy the music

If I would connect a right channel output of a preamp to a 100 Watts monoblock and its right-channel to a 250 Watts one then the right speaker will be 2.5 times louder than the left one. But switching a 100 Watts vacuum tube power amp to a lower power output of TRIODE mode won’t simply make it a 65 Watts one as it still produces the same sound volume (regardless if it is in TRIODE or TETRODE mode)? How then a lower power output of TRIODE mode manifest itself?

My Audio Research Reference 160s is the same way. While the wattage is halved in Triode mode the difference is barely discernible. Careful listening is likely to reveal a bit more slam in Tetrode mode and a bit more warmth in Triode mode. Most of the time you are not using the max wattage. Also, I suspect since the same transformers are used in either mode the current available Is similar. So, keep listening… the difference is really subtle.