Do angels make any requests of G-d besides praising and thanking Him?
Yes Angels make requests of Hashem Zohar Chelek 1, 23,1 see also Pirkei Drabbi Eliezer 13, and Yalkut Shimoni Bereishis 3,1. But they ussually get punished for questioning G-d's decisions by requesting their own way.
דעזא ועזאל הוו מקטרגי עליה, דבזמנא דאמר שכינתא לקב"ה נעשה אדם, אמרו (תהלים קמד ג) מה אדם ותדעהו, מה את בעי למברי אדם, ותדעהו דעתיד למחטי קמך
When the Holy One was making Man, Uzza and Azael the Angels objected that man should not be made as he is destined to sin, "What is man that you should know him." "man is compared to nothing"
שכינתא אמרת לון, בהאי דאתון מקטרגין אתון עתידים למנפל כדכתיב ויראו בני האלהים את בנות האדם כי טובות הנה,
Hashem said to them because of your incrimination against man you yourselves will fall to earth and sin as its written: "the sons of the angels saw the daughters of man and they took them" these were the Nefilim see Nephilim after the Flood
Another example is when Korach and his congregation were undermining Moshe Rabeinu the ministering Angels appointed on the sun and the Moon suggested to avenge Moshe and they moved in protest out of their regular orbit Sanhedrin 110a:
אמר רבא מאי דכתיב (חבקוק ג, יא) שמש ירח עמד זבולה לאור חציך יהלכו מלמד שעלו שמש וירח לזבול אמרו לפניו רבש"ע אם אתה עושה דין לבן עמרם נצא ואם לאו לא נצא עד שזרק בהם חצים אמר להן בכבודי לא מחיתם בכבוד בשר ודם מחיתם והאידנא לא נפקי עד דמחו להו
Why did the sun and moon move to Zevul (level of heaven which is further away) because they protested to Hashem "If you do not do justice for Moshe against Korach and his followers we will go out of our destined places." Hashem then threw missiles at them in punishment and said: " Every day pagan people worship you and you don't protest in my honour, and when a man gets supposed injustice you protest?"
There were many questions about angels' nature on this site - are they real individualities or just G-d's messengers.
To consolidate those views especially on your question, I would like to differentiate between two different aspects of angels - a. the nature of angels and b. our perception of angels.
The nature of angels does not allow for deviating from Hashem's will. They are not separated self-sustained entities. They ARE G-d's will. As I answered about Gabriel they are named after their mission, they have no names for themselves and no "personalities" at all.
The perception of angels in our eyes - as we all tend to personify (anthropomorphism) everything around us, we depict angels as humanlike creature assigning to them all the qualities we humans have. All the interpretations of the angels as separated, self-aware, and self-willing (?) creatures are results of our imaginations, sort of "דברה התורה כלשון בני אדם". Those are parables and not reality, similar to what Rambam writes on the personification of G-d Himself in the Writings (Yesodey Hatora 1).
Based on this differentiation, we can say that the angels really don't have any requests as they have no will of their own, but we picture them as ones that have a sort of free thinking.