Shulhan Aruch (Siman 18) discusses whether bedsheets with four corners are obligated in Sisit. Seemingly, the question is rooted in the Machloket between the Rambam and the Rosh as to what the Gemara means that a "nighttime garment" is exempt from Sisit. The Rambam understands that any garment worn at night is exempt, and therefore the sheets would be exempt at night. However, if one slept wrapped in them during the day, it could be a problem. According to the Rosh, as long as the garment is specifically made for night use, it is exempt even in the day. Therefore, since sheets are made for nighttime use, they would be exempt, even if one continued to sleep after daybreak.
Maran rules that bedsheets are exempt from Sisit, seemingly ignoring the opinion of the Rambam. Some explain that he based his ruling on the Mordechi that says that covering oneself with a sheet does not constitute wearing a garment. Therefore, sheets are exempt, even according to the Rambam. Others explain, from the Eliyah Rabbah, that since he went to sleep at night, when the sheets were exempt, even if he awakens during the day, the sheets remain exempt. The exemption is based on the status of the garment at the beginning of the use. Therefore, even according to Rambam the sheets remain exempt.
Some authorities follow the Magen Avraham (Rabbi Abraham Gombiner, Poland, 1637-1682) who ruled to round one of the corners of the sheet to avoid the problem altogether. He based himself on Rabbenu Tam who clearly stated that covering oneself is also considered wearing. Nevertheless, the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) and the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) understand that the Magen Avraham is referring only to sheets made of wool, which would potentially be obligated MiD’Oraita (By Torah Law). Most of our sheets are made of cotton, but the question would be regarding a wool blanket. Many times a person lies down on the couch on Shabbat afternoon and covers himself with a wool "throw" blanket. There are a number of reasons to be concerned: It is made of wool; it’s meant for daytime use and being used during the day; being covered constitutes wearing according to Rabbenu Tam. Therefore, the English Yalkut Yosef rules that one should round one of the corners.
People often ask when the earliest time for putting on a Tallit is. The Kaf HaHaim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939) writes in Siman 18:18 that the Minhag of Sepharadim in Yerushalayim is one hour before sunrise, which is equivalent to 12 minutes after dawn. These are standard times and apply equally in both the winter and the summer. Hacham Ovadia, following the Pri Megadim, is slightly more lenient and says that one can already don a Tallit six minutes after dawn. Sepharadim should follow one of these two opinions, whereas the Ashkenazim follow the Rema that one may even put the Tallit on at dawn.
Cotton bedsheets are exempt from Sisit. A wool throw-blanket is a potential issue and one of its corners should be rounded.
The earliest time to don a Tallit is 12 minutes after dawn, or even six minutes after dawn