Is there any evidence that Rambam had access to Rashi's works? Did Rashi's works even reach the lands in which the Rambam lived while the Rambam was still alive? Is there any comment of Rambam which, in all likelihood, came from a comment of Rashi?
Did Rashi's works even reach the lands in which the Rambam lived?
To summarize Prof. Shamma Friedman's piece, scholars in the early 20th century assumed from Rambam's silence, that he did not have access to Rashi's works, and furthermore, that they were generally unavailable in Egypt at that time.
Additionally, we can infer from Rambam's early commentaries to parts of the Talmud, that he did not have access to Rashi's works at the time.
However, Rambam's son Rabbenu Avraham clearly had some familiarity with Rashi, as did his Egyptian brother in law, Rabbenu Perachya, and even Rabbenu Chanael Bar Shemuel, Rabbenu Avraham's father in law, a member of Rambam's court in Egypt, had familiarity with Rashi. Rav Yosef Rosh HaSeder, another Egyptian contemporary of Rambam, was also familiar with Rashi.
Was Rambam still alive when Rashi's works reached Egypt?
Given that his younger contemporaries were familiar with Rashi, it is probable that Rambam was still alive when Rashi's works reached Egypt. (Because the only scenario in which he would have been dead at the time, has him dying just before their arrival, providing a very short span of possible time for their arrival, whereas if he were alive when they arrived, there is a much broader possible window for their arrival.)
However, given that Rambam never actually quotes Rashi by name, our only way of knowing whether he was still alive when Rashi's works reached Egypt is by looking for hints of Rashi in the writings of Rambam, which takes us to the last question:
Is there any comment of Rambam which, in all likelihood, came from a comment of Rashi?
If Rambam indeed gained access to Rashi's works later in his life, how could we detect the influence of Rashi? One indicator would be cases in which Rambam changed his view on something to accord with the view of Rashi.
One example of such a change this regards the Mishnah in Bava Metziah (7:2):
ואלו אוכלין מן התורה העושה במחובר לקרקע בשעת גמר מלאכה ובתלוש מן הקרקע עד שלא נגמרה מלאכתו בדבר שגדולו מן הארץ ואלו שאין אוכלין העושה במחובר לקרקע בשעה שאין גמר מלאכה ובתלוש מן הקרקע מאחר שנגמרה מלאכתו ובדבר שאין גדולו מן הארץ
Now, the following [laborers] may eat [of that upon which they are employed] according to scriptural law: he who is engaged upon that which is attached to the soil when its labor is finished, and upon that which is detached from the soil before its labor is completed, providing that it is something that grows from the earth. but the following may not eat: he who is engaged upon that which is attached to the soil. (Trans. Soncino)
This translation follows Rashi (bottom of Bava Metziah 87a) who interprets "עד שלא נגמרה מלאכתו" as meaning that the process of development of the plant is not complete.
However, in his commentary to the Mishnah, Rambam, interprets it as referring to stages of the labor of laborer; not stages of development of the plant. This interpretation is also evident in his Sefer Hamitzvot (267). Similarly, in the first edition of Hilchot Sechirut (12:2), he writes:
והעושה במחובר אינו אוכל אלא כשיגמור מלאכתו אבל בשעת מלאכה הרי הוא בלא תעשה...אלא כל זמן שהוא עוסק בקצירה לא יניף
In these three works Rambam clearly disagrees with Rashi. However, in later editions of this halacha, we find Rambam switching to Rashi's understanding that גמר מלאכה refers to the plant; not the person. His later wording reads:
הפועלין שהן עושין בדבר שגדוליו מן הארץ בין בתלוש בין במחובר, ועדיין לא נגמרה מלאכתו
This term מלאכה now refers to the process of the plant's growth.
In later manuscripts, Rambam completely removes the role of the laborer, fully accepting Rashi's explanation that the only distinction revolves around the plant. He writes:
שהעושה בתלוש אוכל בדבר שלא לאכול נגמרה מלאכתו למעשר ולחלה
In conclusion, there is still no evidence that Rambam was exposed to Rashi's works. However, the degree to which his younger contemporaries were familiar with Rashi's works, is at least sufficient to question the formerly forgone conclusion that Rambam did not see Rashi. At least one interesting change of opinion by Rambam to Rashi's view is certainly suggestive.
 He writes in Shu"t Birkat Avraham (29): ושמעתי שהרב רבנו שלמה הצרפתי ז"ל כך פירש חבטן קצצן אבל לא ראיתי דבריו בזה הענין. Clearly he was familiar with some of his teachings. His comment that he has not seen the Rashi implies that it might have been assumed that he had seen Rashi. His clarification that he has not seen Rashi's words regarding this matter suggest that he has seen other works of Rashi. He also quotes Rashi's Torah commentary, and in HaMaspik (ed. Dana p. 137-8) he quotes the Sefer HaPardes.
 He quotes Rashi by name several times such as in his commentary to Shabbat.
 See for example his commentary to the Rif to Kiddushin, 2b where he quotes Rashi by name.
 He references Rashi's commentaries to Pesachim, Bava Kamm,Avodah Zara, and Chullin.
The Mishnah in Pesachim 4:10 states that Hezekiah hid the Book of Remedies. Rambam in his commentary quotes an explanation of this.
ששמעתי וגם פירשו לי ששלמה חבר ספר רפואות שאם חלה אדם באיזו מחלה שהיא פנה אליו ועשה כמו שהוא אומר ומתרפא וראה חזקיה שלא היו בני אדם בוטחים בה' במחלותיהם אלא על ספר הרפואות עמד וגנזו
I heard, and they also explained to me, that Solomon wrote the Book of Remedies so that if someone would take ill with whatever ailment, he would turn to it and do as it says and be healed. And Hezekiah saw that people were not trusting in Hashem with their ailments, but [were trusting] the Book of Remedies [so] he got up and hid it.
There are some who record that Rashi, in a self imposed exile, passed through Egypt and met the Rambam. The Rambam hosted him and gave him huge honour upon discovering Rashi's wisdom etc.