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Pnei Yehoshua (ad loc.) argues that even had the blessings been Biblical they would still not invalidate a mitzva that is performed without uttering the blessing. The only reason the Gemara states that the blessing is rabbinical is to give an additional reason, but in actuality biblically required blessings would also not invalidate the mitzva.
Regarding the mitzva of Sefirat HaOmer, the Mechaber states (Orach Chayyim 489:7) that if one forgot to bless [and obviously to count as well] all evening long, he is to count during the daytime [hours that follow] without a blessing.
This is so, even though, as the Mishna Berura (ad loc.) notes, there are many poskim who rule that the time to count is in the evening only; nevertheless this would not be considered an interruption of the counting continuity and one then goes on to count each evening with a blessing.
As for your second question, we had a similar query a number of years ago. That correspondent compared the Omer to the Jubilee Year. Because of your question’s timeliness, we will review that discussion.
The mitzva to count the Omer is incumbent upon all men (women are exempt since it is in the category of mitzvat aseh she’hazeman gerama, a positive precept dependent upon time). We are commanded in Parashat Emor (Vayikra 23:15), “U’sefartem lachem mi’mochorat hashabbat miyom havi’achem et omer hatenufa, sheva shabbatot temimot tih’yena – You shall count from the morrow after the Sabbath (i.e., the first day of Passover), from the day when you bring the omer of the wave offering, seven complete weeks shall there be.”
The Mechaber (R. Yosef Caro, author of the Shulchan Aruch) states (Orach Chayyim 489:1) : … [I]t is incumbent (it is a mitzva) upon each and everyone to count by himself; he has to count while standing; he has to recite a blessing before the counting; and he has to count the days as well as the weeks.
The Taz adds that the counting of the Omer is different from the counting toward the Jubilee Year, which is stated in Parashat Behar (Vayikra 25:8), “Vesafarta lecha sheva shabtot shanim, sheva shanim sheva pe’amim … – You shall count [for yourself] seven cycles of sabbatical years, seven years seven times …” The counting of the Omer is also different from the counting of the days toward purification by a person who has become contaminated due to an impure discharge, as stated in Parashat Metzora (ibid. 15:13), “Vechi yit’har hazav mizovo vesafar lo shiv’at yamim letohorato … – When the person … is cleansed, he shall count seven days for his purification …” In these two cases the purpose of the counting is to attain the conclusion of a finite period of time, whereas in Sefirat HaOmer the act of counting itself is a mitzva, and therefore it requires a beracha.
The Magen Avraham notes that we derive that counting the Omer is an obligation incumbent upon each individual from the fact that is is stated, “U’sefartem lachem,” similar to the language used for the commandment to take the Four Species on Sukkot, “U’lekachtem lachem” (Vayikra 23:40).
The Talmud (Menachot 65b) discusses the two verses in Parashat Emor that deal with the counting of the Omer: “You shall count from the morrow after the Sabbath … seven complete weeks shall there be” (Vayikra 23:15), and the verse immediately following (23:16), “Until the morrow of the seventh week shall you count fifty days …” The Gemara concludes that the first verse, using the phrase “seven complete weeks,” refers to the case when the first day of Passover happens to fall on a Sabbath, with the result that the weeks counted are seven full weeks, each starting on a Sunday; whereas the second verse indicates that the counting of the fifty days starts on the second day of Passover - “the morrow after the Sabbath,” meaning the morrow after the [first] day [of rest] of the Festival of Passover - no matter what day it falls on, and the Festival of Shavuot thus occurs when fifty days have been counted. This, indeed, is how we proceed with the counting of the Omer.
The Talmud (ibid.) also cites the pasuk in Parashat Re’eh (Devarim 16:9), “Shiv’a shavuot tispor lach, me’hachel chermesh bakama tachel lispor shiv’a shavuot - Seven weeks shall you count; from such time that the sickle is put to the standing crop shall you begin counting seven weeks.” The Gemara notes that this verse teaches us that the counting depends on [the decision of] the Beth Din. Rashi explains that the Beth Din determines when the holiday is to occur (depending on when the New Moon is seen and attested to before the Beth Din), and we must of necessity ask when Passover begins in order to be able to start counting the Omer on the morrow of the first day of Passover.
How do we arrive at the different interpretations and uses of these pesukim? The answer is to be found in the text itself. Whenever we find the term lachem, the plural form of “you,” the implication is an obligation incumbent upon every individual (and the requirement of a beracha). Whenever the word lach, the singular form of “you,” is used, this indicates that the Beth Din is involved. In the latter case there is thus no specific command of counting. This reasoning applies to counting for the Yovel (where lach is used), since it is the Beth Din’s function to designate the Jubilee Year. As regards the person who has an impure discharge (Vayikra 15:13), counting the days toward purification obviously cannot be the function of the Beth Din (although the singular pronoun lo – or lah – is used) but must be the responsibility of the individual. However, Tosafot (loc. cit, s.v. U’sefartem) point out that since the entire process of counting toward purification can be overturned whenever one sees a fresh discharge, there is no possibility to recite a beracha.
On the other hand, the term lachem refers to the obligation incumbent upon each individual. The Magen Avraham (supra, op.cit.) therefore compares Sefirat HaOmer (U’sefartem lachem) to the mitzva of lulav and etrog on Sukkot, when we are commanded (Vayikra 23:40), “U’lekachtem lachem bayom harishon pri etz hadar … – You shall take for yourselves on the first day the fruit of a citron tree …” Each and every one has to fulfill this mitzva.