Sefirat Haomer

Two ideas permeate our nightly enumeration: 1) The linking up of Pesach & Shavuot & 2) The semi-mourning practices associated with this time of the year, due to the deaths of the students of Rabbi Akiva. Is there a connection?

Each of the 49 intervening days as well as the seven weeks that pass between the holidays is to be counted. It is clear, though, that the Torah is insistent on this count during the interim between these two major holidays of the Jewish calendar.

If one answers an inquiry about what day of the omer it is and does not count again that day, may he count the next day with a beracha? If yes, an onen (before funeral of close relative, who does not perform mitzvot) for a full day of sefira should be able to simulate such a statement and be allowed to continue with a beracha the next day.

Can one fulfill the Mitzva of Sefirat HaOmer by counting while being "half asleep"?

Which is the correct version of counting the omer – "… yamim la’omer" or "… yamim ba’omer"? Is there a content difference or only a grammatical one between them?