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Does a shul or Beis Midrash need to be checked for Chametz?
Shuls and Batei Midrashim need to be checked on the night of the 14th with a candle. The Gabbaim of the shul are responsible to search the shul for Chametz. The Gabbaim may say a blessing on their search for Chametz in the Shul and Beis Midrash. The Gabbaim do not have to nullify the Chametz after completing the search. [However, Shuls who provide their own Chametz, such as by Kiddushim, should nullify the Chametz after the search.] If the Gabbai transgressed and did not check the Shul at night of the 14th, then he may Lechatchilah check it on the day of the 14th using sunlight, without a candle, if the Shul has many windows.
The Gabbai of the shul is responsible to search the shul for Chametz with a blessing on the night of the 14th.
Does the Gabbai say a blessing both in the shul and in his home?
Yes. A blessing is to be said in both areas, whether he first checked the Shul and then his home, or vice versa. Some Poskim however suggest that the Gabbai should have in mind while saying the blessing for searching his house to also include the Shul which he will check later on.
Should the Gabbai sell the Chametz of the Shul?
The custom is for the Gabbai, or another person appointed by the shul directors, to sell the Chametz of the Shul. This, however, does not exempt one from the obligation of searching the Shul for Chametz.
 Admur 433:36
 The Gabbaim are not careful to search it at night, rather they suffice with sweeping it well during the day. However, this is not proper and they need to be warned against this and to check [the areas at night] as the sages instituted. [Admur ibid]
The reason: This applies even if there is no food ever eaten in the shul being that children enter Chametz into them throughout the year. [Admur ibid; Certainly it applies if Kiddushim and Farbrengens are eaten there throughout the year.]
 Admur ibid; M”B 433
Other Opinions: Some Poskim rule a blessing should not be said. [Aruch Hashulchan 433]
 Admur ibid; Peri Chadash 433; Mishneh Berurah 433; See however Peri Megadim 433 A”A 19
The reason: Being that they cannot nullify and disown Chametz which does not belong to them, as a person cannot disown something that is not his. [Admur ibid]
 Pashut! The above exemption was only because they don’t own the children’s Chametz.
 Admur 433:37
The reason: Being that it is common to have many windows in a shul, and they thus have a lot of light. [Admur ibid]
 Implication of Stimas Haposkim, as it is two separate Mitzvos, and changing areas is a Hefsek which requires a new blessing.
 Daas Torah 433
 Piskeiy Teshuvos 448:2; Teshuvos Vehanhagos 1:287, brought in Piskeiy Teshuvos 433:6; Nitei Gavriel
Ideally, the Gabbai is unable to sell the Chametz of the Shul, just like he may not nullify the Chametz, being that it does not belong to him but rather to others. However today that the Shuls themselves provide Chametz, and hence contain their own Chametz, there is room to say that they should sell the Chametz of the Shul.