The Tur (581) writes that one should bathe himself and take a haircut before Rosh Hashanah based on the Medrash which states that one should eat, drink and rejoice on Rosh Hashanah since he knows that Hashem will perform a miracle for him.
The Yom Teruah asks from our Gemora which states that song is not recited on Rosh Hashanah. He answers that each individual can rejoice since he is confident that Hashem will judge him favorably however there will be those in the world that will be sentenced for death and because of them, hallel is not recited on Rosh Hashanah.
The Aruch Lener comments that the Heavenly angels did not inquire about themselves since they understand that they should not sing when there are people being inscribed for death. They were only asking in regards to Klal Yisroel, that they should recite hallel since they should be secure that they will be inscribed for a good year. Hashem answers that nevertheless, it would be inappropriate for them to recite hallel cheerfully while they are being judged.
He writes further that the Heavenly angels were asking about Klal Yisroel reciting hallel since they do not have permission to sing praise to Hashem unless Klal Yisroel sings first, so in essence they were asking regarding themselves.
The assumption of the Aruch Lener that the heavenly angels refrain from singing on Rosh Hashanah is not universally accepted. Tosfos in Eruchin (10b) writes explicitly that the Heavenly angels do sing on Rosh Hashanah. His proof is from our Gemora that the angels do not inquire about themselves. It is evident that they do sing and their question is only regarding Klal Yisroel.
Reb Yonason Eibshitz in Yaaros Devash (14) explains that the Heavenly angels claimed that while they understand why Klal Yisroel does not recite hallel on Rosh Hashanah but they should be able to sing since there is no book of death opened for them. The answer given to them is that there is a judgment for them as well (like it is said in the tefillah of u’nesaneh tokef). The book of life is referring to the judgment for the angels since they live forever. The book of death is opened for people since they can be inscribed to die. This is the reason why the angels and Klal Yisroel refrain from singing on Rosh Hashanah.
The Maharsha in Eruchin asks why we are able to recite ‘az yashir’ on Rosh Hashanah. He answers that this is only relating what Moshe and Klal Yisroel sang when they crossed over the sea.
It is cited in the name of Reb Chaim Brisker that it is permitted to recite the shir shel yom on Rosh Hashanah since that does not require complete happiness when it is being recited. Hallel can only be recited if one is in a state of complete happiness and that is not possible on Rosh Hashanah.