What are the origins of the expected appearance of Elijah on Passover and does it also extend to the Messiah?

Many old Pesach Haggadahs, starting about 550 years ago, began drawing Eliyahu (pictured blowing a shofar, announcing the Mashiach, see Malachi 3, 23) and Mashiach (pictured riding on a donkey, Zecharia 9, 9) on the page of the beginning of Hallel when we open the door after the meal to invite the redemption (and some had the custom of actually "inviting" Eliyahu and the Mashiach). This is directly connected with the opinion already found 1700 years ago (Sanhedrin 98a) that just as the redemption from Egypt was in this month of Nissan, so too will be the future and final redemption. This is also connected with the difference of opinion (Peshachim 118a) whether to drink a 5th cup of wine at the Seder. What we do is a compromise, where we pour the 5th cup (together with the 4th cup of Halllel) but don't drink it. Some connect this with the 5th language of redemption, for we drink the 4 cups in memory of "and I removed", "and I saved", "and I redeemed", "and I took out", (Talmud Yerushalmi Psachim 10, 1, based upon Shmot 6, 6-7) but there is also "and I will bring you to the Land" (ibid, 8), which hasn't yet been permanently fulfilled, so we look forward to the Messianic age when it will be totally and eternally fulfilled. Accordingly, many refer to this 5th cup as the "Cup of Elijah", who announces the Mashiach is coming, and then we will drink that 5th cup.