1.  From Moses' birth onwards through the end of the Torah (Pentateuch), Moses' name appears in every single parashah (weekly portion) except one. What event occurs in the week that parashah is read?
Moses' yahrzeit (the anniversary of the death of Moses).  

2.  How many aliyot are given on the intermediate days of Festivals?
Four (not counting Maftir).

3.  What is the only congregational service that does not include an Amidah?
S'lichot

4.  If one has a doubt whether or not one recited a blessing one should not recite it again. What is the exception?
Birkat HaMazon (The Blessing for Sustenance, said after meals). C.f., Kitzur Shulchan Aruch 6:6 & 44:11.

5.  Name a sacrificial service NOT displaced by an Amidah.
The Pascal (Passover) offering.

6.  The requirement for Torah-Reading is due to Rabbinic Enactment rather than Torah Law. Which two parashah are read (according to many authorities) in fulfillment of biblical mandate?
Parshat Zachor & Parshat Parah

7.  What is the most infrequent time-dependent Jewish liturgical event?
Birkat Ha-Chamah, once every 28 years.

8.  When is Shir HaShirim (The Song of Songs) read publicly in a synagogue?
The Sabbath of Passover.

9.  What is the only Biblical event dated according to the passage of years from the Exodus?
Erecting the Temple.

10. When is Mizmor L'Todah (Psalm 100) omitted from Psukei d'Zimrah (The Verses of Praise) on a weekday, and why?
This Psalm corresponds to the ancient Thanks-Offering, which had to be eaten by midnight the same day donated. So it is not said on erev Yom Kippur, nor on eruv and chol hamoed Pesach -- because it was chametz.