How exactly, did Speed get accepted into the Racing Academy in the first place, especially since he lacked his own car, which was apparently a requirement?
Perhaps the car he was going to use broke down or was stolen along the way, and, confusing Racing Academy with Alpha Academy (which actually does lend out race cars to students)he decided that it wasn't a problem? As to how he got there, perhaps he won a scholarship or took out a loan?
So Speed, the most famous race driver on Earth, up and vanishes, but somehow had the time beforehand to spawn a second kid and stick him in an orphanage with a couple of Plot Coupons?
The show seemed time and again to reveal the reasons for all this happening, but as Lucy put it, "It feels like we've got more questions and no answers." They obviously want to keep the episodic feel for as long as possible.
And Speed Jr. could have been unplanned...
Why, exactly, wasn't the Mach 5 on display in a museum somewhere?
No one knew what happened to the Mach 5 until Speed Jr. discovered it in pieces in a junkyard near the school.
After for how it wound up in that junkyard, the Mach 5 could've been replaced with another more advanced car a few years after the anime ended, and its legacy wasn't as long-lived as the Racer family thought. Technology is always evolving anyway. Or, it may have been in so many chases that it was worn out beyond repair.
The implication is that it was apparently destroyed the same time Speed Sr. was apparently killed. As for how it ended up in easily-salvaged pieces in the school junkyard, the smart money would be on Speed Sr. putting them there.
Spritle has a last name. Why does no one use it?
Perhaps Spritle just prefers to be called by his first name, possibly because the look on people's faces when they say 'Spritle' (especially when it's the name of their boss) is just so darn funny!
That would be just like the little brat.
We can assume Pops, Mrs. Racer and Inspector Detector all passed on years ago. So... What happened to Trixie and Racer X?
We don't know that those three all passed away because the show didn't say otherwise. The whereabouts of these characters are unknown.
The show is set 40 years after Speed Racer do you really think Pops would still be alive? Or at least spry?
Considering Pops was a heavy smoker, and in constant danger due to his son's wild shenanigans involving gang members and thugs, I think it's safe to say that he's passed on, although X does mention his grandmother once.
Rex could have died in the line of duty. Or he and Trixie could have died in the same events that caused Speed to be presumed dead. As for Mom and Pops, it's entirely possible that the writers didn't miss that Speed was Japanese and as such the elderly couple are still in their home country.
Actually, Trixie went nuts 16 years prior to the show after an accident with her father and the Mach 6...and became the Big Bad of show.