Is it permissible to compose a specific supplication that includes thanksgiving, praise, prayers upon the Prophet, and all the supplications mentioned in the Sunnah, along with mentioning all one's needs, to be recited daily and at times when supplications are answered?

There is no harm in collecting the supplications (du'as) mentioned in the Sunnah and supplicating with them. They are better and more blessed than others. It is sunnah when supplicating to begin by praising and extolling Allah, then sending blessings upon the Prophet (peace be upon him), and then supplicating with whatever one wishes. The origin of the athkar (remembrances) and du'as that are athur (transmitted) is Tawqif (that they are fixed and not subject to human alteration). Therefore, one should preserve their wording and number without addition, subtraction, or alteration. The supplications mentioned in the Qur'an and Sunnah are those that are legislated to adhere to, memorize, and spread. As for others, they may be permissible or contain ambiguous phrases. As for the transmitted supplication from the Prophet (peace be upon him), if it is not obligatory, the servant has the choice to supplicate with all of it, or some of it, or to leave it. However, if it is obligatory or a confirmed sunnah, such as seeking refuge from the punishment of Hell and the punishment of the grave in the final tashahhud, then it should be performed.