כי שמחתני ה' בפעלך במעשי ידיך ארנן (ב"מזמור שיר ליום שבת")
This verse contains past tense (שמחתני) — ‘That You, HaShem, have gladdened me with Your deeds’ and the future (ארנן) — ‘I will sing for you at the work of Your Hands’ (reference is to the Bais HaMikdash).
It is possible to explain that the בפעלך is not a reference to actions directly by G-d, but actions of others (the Jews) done at His direction. Many places the activity, though done by others, is attributed to the one that directs the action (simple example, the activities of all the cooks are attributed to the master chef who directs all the actions) שליח של אדם כמותו.
Similarly, Tosfos (Menachos 21b) understands the statement of R’ Yochanan ben Zakkai (in reference to the Parah Adumah) that he calls the Parah Adumah “my work and service”. Yet R’ Yochanan was not a Cohen, but this means that the procedure was done at his command, and thus, was described as if it was his work and service.
Thus, the verse means that the actions of building the Bais HaMikdash were done by the hands of man, but are attributed to the Commander.
However, the words מעשי ידיך could only refer to the actions of HaShem. For example, Tehilim 19, 2, ומעשה ידיו מגיד הרקיע refers to HaShem, since at the time of Creation, no one else existed. Another example is the famous Midrash that when the Egyptians were drowned by HaShem, He told the angels not to say Shirah, since מעשי ידי were drowning.
Therefore, the first part of the verse (past tense) refers to the first two Batei Mikdashim that were built by others at HaShem’s direction. Whereas the second part of the verse refers to the Third Temple, which is built completely by the Hands of HaShem. Thus, we express our joy at the work already done of building the first Temples (through the deeds of the Jews at HaShem’s direction) but with the deeds of Your (direct) Hands, the building of the Third Temple (which will be built by HaShem) we will then sing in the future at the time of its revelation (speedily in our days).
ותתן לנו את יום הכפורים הזה
Why is the word (כפורים) plural and not singular like the other descriptions of the day (יום סליחה, יום מחילה)?
Perhaps, one can explain by understanding that Teshuvah can function in steps, and thus the forgiveness can be in multiple levels (thus plural word כפורים).
This concept of multiple levels is symbolized by the description of white; sometimes the forgiveness is described by white as snow (solid color, a complete forgiveness) and other times by white as wool (a less bold color of white, an incomplete forgiveness).
Similarly, the “size” of the sin can be expressed by different levels of the color red, crimson (blood) red or a duller shade of red. The amount of heating of the blood (passion of feeling) involved in the sin is reflected in the boldness of the color of red.
In Parshas Chukas, Rashi (19,22) explains the reason for the name Parah Adumah is due to this offering comes for atonement of the sin (of the Calf), and sin is called אדום red. However, it does not explain why the color is used to indicate this idea.
Now, we can understand red since this concept of that color reflects levels of involvement and effort in doing wrong as reflected of the redness in the face. And the sin of the Calf was due to great effort. Thus, this sacrifice utilizes the deep color of red in order to bring the level of forgiveness that is needed to overcome the great deal of effort involved in this sin.