In that thread, there is a reference to hauling a piano up to a third story apartment. When the rope breaks, a man below the piano on the other side of the building is safe, but if he is under the piano, he is squashed. In English, there is a difference between below and under.
The difference is a little subtle:
if you are under the piano, you are right underneath it, so if it falls, it will squash you;
if you are beneath the piano, you are merely at a lower point in space, which may include being directly under the piano but may not be.
This site give a good explanation, I think. http://www.bbc.co.uk/worldservice/learningenglish/radio/specials/1535_questionanswer/page35.shtml
"Bajo la mesa" = "below/underneath" the table"; "bajo el coche" = "below/underneath the car" "la mesa baja" = "the short table"; "el chico bajo" = "the short boy" "bajo el precio" = "I lower the price"; "baja el puente" = lower the bridge" Used as a preposition, it is always "bajo". Used as an adjective (as you thought), it is "bajo/baja"; whichever is appropriate to the gender of the noun Used as a verb, it has all the forms of a regular verb.
How can you tell which to use? By the magic of context.
bajo (la mesa) - is not a position/placement/location of something:
dinero bajo la mesa - money under the table (not actually underneath)
bajo su dirección - under his dirrection
When speaking position/placement/location of 'de' comes into play:
abajo de la mesa - below the table debajo de la mesa - under(neath) the table
May as well. "Under the table" generally implies something being crooked, or at least kept out of sight. Being paid under the table implies something shady, while laying your cards on (top of) the table means holding nothing back. If nothing else,it's good for a joke which may help us remember the phrase.
So,, to return to the topic, does "bajo la mesa" have the same shady connotations en español?
No. There are a lot of instances where English uses an idiomatic expression (one that means something quite different from the literal meaning of the words) and Spanish also has an idiomatic expression, but they are not the same. For example, something in English may cost "an arm and a leg", but in Spanish, it will cost "an eye out of your face".
In all honesty... I don't care whether you ask me "What do you have under the table" or "What is under the table." But let's be real... We would all ask "What is under the table." Aannd obviously in an apprehensive tone because who would ask that type of question without a hint of anxiety in their voice.