I heard that when setting up a ping-pong table on Shabbos one must reverse the order, i.e. first hold the board in the air and then position the legs, is this true? 

No, it is false and the following will explain why.
The Gemora in Beitza 32b teaches us that certain keilim when erected on Shabbos must be erected in a backhanded manner, i.e. different to the way it is normally set up. This is because when erecting these particular keilim it appears as if one is making a tent – first the legs and then the roof or cover – and therefore a variation from the norm is required.
However, this only applies to keilim whose function is similar to a tent’s. Just as a tent’s inside is utilized for living in, usage etc, so to keilim whose ‘inside’ or ‘underside’ is used require a deviation when setting up.
A table, Tosefos 1 tell us, is not an item whose underside is used and therefore it would not require deviation from the norm and may be set up in the regular manner.
Therefore when setting up a ping pong table one may first position the legs and then place the board on top of the legs.  

Do you have an example of a k’li where the deviation from the norm is necessary? 

The gemora tells us that one uses the space beneath a bed and the Rishonim say that it is used for storing shoes or sandals during the night. (Food is not stored beneath a bed because an unclean ruach descends upon the food). However, Tosefos says that although the underside is used, since it is not the primary reason for setting up the bed, one need not change from the norm unless the bed has two sides (headrest and footrest for example) that either reach the floor or are within three tefachim 2 from the floor.

In such a case one must first hold the board in the air and then slide the legs in underneath. Since it is setup in an awkward manner one will remember that tent erecting on Shabbos is forbidden and will not erect a tent.

What about setting up a bench on Shabbos? 

It has the same halacha as a table, because a bench’s underside is not used either.

What is the halacha regarding the use of barrels as table legs when the barrel is being covered by the board?

This issue is based on the previous one. The Taz says that covering a four-sided k’li automatically accredits it as a usable space and requires a deviation from the norm. Since a barrel has four sides, when using it as table legs one must hold the board in place and then slide in the barrel.

The Shulchan Aruch HaRav 3 disagrees with the Taz and permits erecting a table even when it has four sides but the Mishna Berura 4 paskens like the Taz.

Does it make a difference whether the legs were already in position?

A crucial point. The Mishna Berura 5 says that one is only required to vary from the norm when one first positions the legs and then places the top but if the legs were already in position and one is merely placing the top, no deviation is necessary.

This is because Chazal required a deviation from tent building, which is usually accomplished by positioning the sides and placing the roof. In this case it does not resemble tent building and hence no need for the deviation.

Is there a problem covering a wide bath or tub on Shabbos?

In yeshiva there used to be a large, wide tub on wheels which was used for storing the bread and challa used for Shabbos. The Shulchan Aruch 6 says that when covering a large tub one must make sure that he does not completely cover the tub. Another way around the problem is if the contents of the tub reach the top, because an ohel is fashioned when there a space of at least a tefach³ (squared) is covered, but when the contents reach the top an ohel is not formed, thereby sidestepping the entire issue.