It is very hot at night and I sleep without clothes on. Is it permitted to recite the bedtime shema and hamapil unclothed if one covers himself with his bedsheet? Should he also wear a kippa ?

ב"ה לע"נ אברהם יוסף בן אלתר משה ז"ל Shalom The first issue to be addressed is one allowed to leave himself totally unclothed and uncovered. According to the Shulchan Aruch (או"ח ב, שו"ע הרב סי' ב) this is improper since a person should not say that he is in the inner most room of his home and who can see me? Since Hashem's greatness is all over the world. The Mishna Brura explains though, that when one is unclothed partially in the toilet or fully when showering or bathing in a river [or the like] , is done out of necessity this is not to be considered immoral. Under a sheet you may be unclothed, however when saying Shema covering yourself with a sheet is not enough unless the sheet is held against you tightly above your waist . ( שולחן ערוך אורח חיים סי' ע"ד, משנ"ב ס"ק א) Also, even while under a sheet your legs should not be touching your genitals while saying Kriyat Shema. (שו"ע או"ח עד:ה) When saying Hashem's name one must have a kippa on his head. (שו"ע או"ח סי' ב:ו,משנ"ב ס"ק יב, צא:ג) All the best and have a pleasant summer.