Answers aren't coming quite as fast and furious as in previous episodes -- likely due to the incredible unfairness of my selections (heheheh) -- so I'm going to drop some super-detailed clues for you guys. Let's see if you can figure anything else out from these!
01: This is the first freeware game ever to appear on one of my lists, and it only really qualifies for the list because despite being freeware, it's available through GOG, making it a "free retail game." An updated version of it (that will NOT be freeware) is in development now. (It's also REALLY good, so once you find out what it is, I encourage you to look it up and check it out -- it won't cost you anything to try, and it may wind up becoming a favorite!)
02: This PS2 classic has two opening scenes: one leading into the title screen, and another "attract mode"-style movie that plays if you sit on the title screen for a while without pressing anything. This snippet is taken from the latter.
03: This late-life SNES title was heavily inspired by (and apparently was originally going to be a licensed adaption of) an insanely popular '80s franchise that has recently been resurrected with unfortunate results. It's a technological marvel for the system, but its high level of difficulty and extremely complicated control scheme ultimately kept it from being regarded as one of the SNES greats.
04: This is one of the seminal Vita experiences, and was a system seller for a lot of people.
05: This Wii title is a popular entry in a popular series that originally spawned off of another popular series.
06: At the time of its release, this game was supposed to be Sony's answer to a popular Nintendo franchise. It didn't quite live up to that reputation, but it did OK for itself anyway.
07: This comes from a game widely considered to be one of the best on its entire system. It has a strong connection to the date 1997 October 1.
08: This is from a Falcom title.
09: This game was released on PS2, PC and PSP. Although all three versions are graphically and functionally more or less identical, the PSP version is generally considered the black sheep because it was re-dubbed from the original PS2 and PC versions, losing a bit of the original English dub's charm in the process. For some reason, this opening theme was arranged differently for the PSP version as well, though the melody remains the same (and is the main theme for the entire game).
10: This is the most correctly-identified track on the entire list so far, so you don't get a clue for this one. Either you recognize it, or you don't.
11: This game used to be readily available through digital channels, but was sadly pulled from them due to licensing issues. It's now fairly hard to come by.
12: The soundtrack to this game was composed by a man who used to be a famous MOD tracker back in the Amiga's heyday. One of his award-winning MODs, in fact, was arranged and used in the game's soundtrack as a throwback to his early successes. This opening theme, too, was not only written by him, but performed by his band, which has numerous published albums and is quite successful in Europe.
13: This is generally considered to be one of the PSP's last big releases.
14: This Japan-only story-driven RPG for the PS2 takes a lot of gameplay cues from Sakura Taisen, and while it's not a particularly well-remembered game, it has quite a pedigree behind it, having been developed by the same team responsible for two Japan-only RPG classics from the 32-bit era, one of which was one of the earliest games in the country to be censored in some of its releases due to intense graphic violence and gore.
15: This franchise RPG was widely publicized for featuring both normal and Hatsune Miku versions of every song on its soundtrack, all penned by a very well-known game music composer. The one exception was this opening theme, which only exists as you hear it here, and which was composed specifically for this game by a renowned Vocaloid musician.
16: It's highly likely you've never heard of this PS1 action RPG, so here are a few clues to help you narrow down the list: it's available on Japanese PSN as a PS1 classic, it was covered in an import review by RPGamer, and although it is a great game (a REALLY great game, in fact!), its biggest claim to fame is probably its music plagiarism, with tracks that blatantly rip off the Dragnet theme, as well as (quite randomly) the 1970s Delfonics ballad "La La (Means I Love You)."
17: This track comes from a relatively early Gust title.
18: Most people think the Roman numeral in this game's title represents its position in the series -- and indeed, counting unnumbered spinoff(s), said numeral does match up with its release order quite well. However, technically speaking, this is a series spinoff in and of itself, NOT a numbered entry -- as evidenced by the fact that the next numbered entry in the franchise contained a number that was actually one LOWER than the number in this game's title.
19: This virtually unknown Dreamcast game features a soundtrack composed not by a renowned game musician, but by a renowned Japanese new age artist. He's done other video game music before and since, including for a previous entry in the series #18 is part of, but this is a bit of an interesting case since the game's soundtrack doubles as one of his album releases.
20: This, too, is from a Falcom title.