A DOUBT ABOUT COUNTING SEFIRAT HAOMER
If a person is in doubt if s/he counted Sefirat HaOmer on the previous day, the Terumat Hadeshen  writes that s/he should continue to count with a blessing, and the Shulchan Aruch agrees [Shulchan Aruch, Orach Chaim 489:8]. According to the Pri Chadash, the reason is that there is a double doubt ["s'fek s'feka"]: It is doubtful if he counted or not, and even if he did not count, there is a doubt whether to accept the ruling of the Tosafot, to continue with a blessing even after missing one day. This was also quoted by the Mishna Berura [ibid. note 38], and the Pri Megadim uses this to show that a blessing can be recited in the case of a double doubt [Shulchan Aruch, Orach Chaim 143:5].
The halacha can be summarized as follows:
(1) If one forgets to count at night, s/he can count during the following day without a blessing, and then continue the remaining nights with a blessing.
(2) One who forgets at night and the following day should continue to count without reciting a blessing.
(3) One who is not sure whether s/he counted (or might have made a mistake in the count) can continue to count with a blessing.
SEFIRAT HAOMER and SHE'HECHIYANU
While there are authorities who forbid purchasing food or items during sefira that would require a blessing of she’he’chiyanu [Eliyahu Zuta 493:1; see further, Natei Gavriel Pesach vol. 3 54:1]; many others permit reciting this bracha and therefore allow people to purchase new clothes and other such items [Mishna Brurah 493:2; Rivevot Ephraim 8:227; Halichot Shlomo - Moadim vol. 2, 11:14 note 53].
Furthermore, even according to those who are more stringent:
1. Reciting she’he’chiyanu on Shabbat, Yom Tov, Chol HaMoed and Lag Ba’Omer (and other days when the mourning of sefira no longer applies) is permitted [see Tzitz Eliezer 18:41 B’tzel HaChochma 4:48].
2. In the case of need, there are those who permit buying and wearing new clothes [Yabia Omer Orech Chaim 3:26, Yechaveh Daat 1:24].
3. Purchasing a new home, buying furniture, painting or remodeling is permitted [Tzitz Eliezer 18:41:3, Nitei Gavriel Pesach vol. 3 59:4-5; Iggrot Moshe Orech Chaim 3:82].
SEFIRAT SHAVING DURING HAOMER
According to the Shulchan Aruch, there are two components to the observance of Sefirat HaOmer. The first is the actual counting of the Omer, something which should be done each night between Pesach and Shavuot.
However, the second component of Sefirat HaOmer, is the period of mourning that we observe during this time. While there are many different customs related to exact dates on which we must mourn, during this period, we have accepted the custom not to perform weddings, not to dance or play live or recorded music and not to get haircuts or shave.
Yet, on this final point, there is some debate among contemporary poskim, especially for men who shave on a daily basis.
According to Rabbi Moshe Feinstein, if not shaving would lead to significant problems in the business environment, a person is allowed to shave during Sefira [Igrot Moshe Choshen Mishpat 1:93].
According to Rabbi Shlomo Zalman Braun, it is permitted to shave on Fridays in preparation for Shabbat [Shearim Ha Mitzuyanim B'halacha 120 note 5].
According to Rabbi Aharon Lichtenstein, since the level of mourning during Sefira is analogous to the mourning period following shloshim (i.e. "yud bet chodesh), and because the honor due to Shabbat takes precedence over the mourning of Sefira [Taanit 26b], therefore it is an obligation for those who regular shave, to shave for Shabbat during Sefira.
According to Rabbi Yosef Dov Soloveitchik, and Rabbi Ahron Soloveichik, since a day-old beard for a person who shaves daily is considered unkempt ("Soloveichik, since a day-old beard for a person who shaves daily, is considered unkempt ("k'dei she'yigaaru bo chavereirav"), a person is allowed to shave every day of Sefira. [Nefesh HaRav p. 191].