I have college coming up in 3 days and I already shared this first part with Flepty, so I might as well share it with the rest of the community.
In Sports Mix, the sports you get for each of the matches against the Expert AI are completely randomized. This means the results you can get are split into four categories: Basketball First (or BBall First for short), Volleyball First (or Volley First for short), Dodgeball First and Hockey First. Now, Hockey First lineups are commonly dreaded amongst the Mario Sports Mix speedrunning community, but that’s something I’ll discuss much later on.
The lineups are as follows (note: due to the Basketball Expert AI in Mario Sports Mix being unfathomably hard, getting anyone of these six lineups is horrible unless you’re extremely good at the game):
 Round 1 Basketball, Round 2 Volleyball, Round 3 Dodgeball
 Round 1 Basketball, Round 2 Volleyball, Round 3 Hockey
 Round 1 Basketball, Round 2 Dodgeball, Round 3 Volleyball
 Round 1 Basketball, Round 2 Dodgeball, Round 3 Hockey
 Round 1 Basketball, Round 2 Hockey, Round 3 Volleyball
 Round 1 Basketball, Round 2 Hockey, Round 3 Dodgeball
(note: from this point on, these lineups will have Basketball in the position of either Round 2 or Round 3. Regardless, any lineup for Sports Mix that includes Basketball can be a runkiller, especially in the Star Cup):

Round 1 Volleyball, Round 2 Basketball, Round 3 Dodgeball

Round 1 Volleyball, Round 2 Basketball, Round 3 Hockey

Round 1 Volleyball, Round 2 Dodgeball, Round 3 Basketball

Round 1 Volleyball, Round 2 Dodgeball, Round 3 Hockey

Round 1 Volleyball, Round 2 Hockey, Round 3 Basketball

Round 1 Volleyball, Round 2 Hockey, Round 3 Dodgeball (featured in Zojalyx's 1:27:26 & Flepty's 1:01:04)

Round 1 Dodgeball, Round 2 Basketball, Round 3 Volleyball (beneficial on Flower Cup for skipping Daisy Garden)

Round 1 Dodgeball, Round 2 Basketball, Round 3 Hockey

Round 1 Dodgeball, Round 2 Volleyball, Round 1 Basketball

Round 1 Dodgeball, Round 2 Volleyball, Round 3 Hockey

Round 1 Dodgeball, Round 2 Hockey, Round 3 Basketball

Round 1 Dodgeball, Round 2 Hockey, Round 3 Volleyball

Round 1 Hockey, Round 2 Basketball, Round 3 Volleyball (beneficial on Flower Cup for what I like to call WJ3, or Western Junction 3)

Round 1 Hockey, Round 2 Basketball, Round 3 Dodgeball

Round 1 Hockey, Round 2 Volleyball, Round 3 Basketball

Round 1 Hockey, Round 2 Volleyball, Round 3 Dodgeball (featured in Circinus' 1:05:48)

Round 1 Hockey, Round 2 Dodgeball, Round 3 Basketball

Round 1 Hockey, Round 2 Dodgeball, Round 3 Volleyball (featured in Zojalyx's 1:27:26)
This means there are 24 possible lineups for a Sports Mix tournament (not counting the Behemoth King battle at the end of Star Cup), and 18 of them have Basketball as one of the sports (with 6 of them even having it as the first sport of the lineup). As a result, the odds needed to avoid Basketball in a Sports Mix tourney entirely are 1 in 12. However, that’s only just the beginning.
See, in order to get a better grasp of how RNG in a Sports Mix Tournament works, we need to dig a little deeper. Yeah, we have the lineups to go off of, but that’s barely even scratching the surface. Let’s discuss the odds of avoiding Basketball in just one Sports Mix match, which are 1 in 4.
Now, these may seem like easy odds, as each of the lineups I described only have one listed first sport  hence the naming of said lineups within the speedrunning community. However, when speedrunning Sports Mix (regardless of whether you’re doing All Cups or not), these need to be kept track of. Outside of what I like to call “The Six Saviors,” any other lineup can kill your run  especially the Basketball First lineups  if you don't know what I like to call "Shot Clock Abuse". And this is only just the tip of the iceberg.
Let’s get back to those 1 in 12 odds I discussed earlier. As the 1 Match odds for no Basketball are ¼, or 25%, that means those odds are multiplied by ⅓ to get the 1 Tournament odds for no Basketball, which are ¹⁄₁₂  an 8% chance (I used 3 for the denominator as there are three rounds in every tournament).
(continued)
But what if we were to take those 1 in 12 odds and multiply those by ⅓? Well, that’s what I did… and I was actually very surprised by the results. The odds of avoiding Basketball in EVERY Sports Mix Tournament is 1 in 36. A whopping 3% chance if rounded.
Those wanting to speedrun each of the Sports Mix Cups separately are already in for a rough ride as it is, since they only need to succeed in getting 1 in 12 odds on just a single cup. But if someone wants to do a perfect All Cups run for themselves, they need to…
 Succeed in getting the ¹⁄₁₂ odds on Mushroom Cup.
 Succeed in getting the ¹⁄₁₂ odds again on Flower Cup.
 By some miracle, pull off getting the ¹⁄₁₂ odds a third time on Star Cup.
 Round off the run by not messing up Team Final Fantasy and/or Behemoth King.
The chances of this run flopping are ³⁵⁄₃₆, which is 97% if put into decimals. With all these painstakingly high odds & brutal AI stacked up against a mere speedrunner, there’s only one question left to be asked… is the run even possible?
Well, when Sports Mix Tournaments have broken AI that will annihilate you in Basketball, insurmountable odds that have to be overcome through luck alone, painful runkillers, and the hell that is RNG manipulation, then most people would be quick to assume that a dead end has been reached.
However, having had some experience with this game in the past, I decided to look through Flepty's 1:01:04 and see what a theoretical perfect run (3 cups with Volleyball, Dodgeball and Hockey) would look like by adding up his Volleyball, Dodgeball & Hockey match times.
As it turns out, doing this led to the conclusion that the No Behemoth runs can go as low as 49:15, while the runs where you do fight Behemoth King can go as low as 53:46.
Going back to the question, the answer I have to offer is that this run can be humanly possible. However, you need absurdly good luck on top of very good execution. Plus, the Expert AI and Behemoth King can only serve to make this even harder to achieve.
Moving onto the other little bits, I found out you can build up the Special Meter in Volleyball by using the gimmicks on both Peach's Castle and Luigi's Mansion. Building it up in Peach's Castle involves using the sprinklers, which pop up in 6 different patterns  each popping up at random, though some are more common than others.
 Left Side Blocked (meaning the Center & Right areas are left open for shots)
 Right Side Blocked (meaning the Center & Left areas are left open for shots)
 Left & Right Sides Blocked (meaning only the Center area is left open for shots)
 Left to Right (meaning depending on where you're aiming & depending on how much time has passed, you'll have less time to pull off landing a shot aimed to the Left area.)
 Right to Left (meaning depending on where you're aiming & depending on how much time has passed, you'll have less time to pull off landing a shot aimed to the Right area.)
 Center Blocked (meaning the Left & Right areas are left open for shots)
Luigi's Mansion's method of doing this is much easier, though  again  it is luck based. If a Gold Ghost pops up on the opponent's side of the court, you can do another set of volleys by having the ghost intercept your shot. You can do this multiple times in order to max the meter out so you can use your Special Shot. Depending on how hard the Gold Ghost spikes the volleyball in order to intercept, this can be easy or hard. It can either do one of the three following things if you launch the volleyball towards it:
 A block, the best case scenario.
 A soft spike.
 Or a hard spike, the worst case scenario as you lose a bunch of time due to either you or your teammate tumbling backwards after receiving the ball.
The only downside to these I could find is that your opponent can also benefit from the Peach's Castle & Luigi's Mansion exploits, as they can also max the meter out by abusing the sprinklers or Gold Ghost respectively. That's all I have to say for now. Thanks for your time.
I'm sorry but I am gonna quickly correct your odds.
To clear up, the odds of NOT getting basketball in a single Sports Mix match are 3/4, not 1/4. This is beacuse out of the 4 random options, only 1 is basketball, hence 1/4 to play basketball, and 3/4 to not.
The odds of having a basketballfree whole cup is 1/4 instead. The easy way to explain this is:
Out of the 4 sports, exactly 1 will not be played per cup, and this is random. Therefore the chances of it being Basketball is 1/4.
The maths way is to multiply the odds of no basketball at Round 1, 2 and the final. This is 3/4 x 2/3 x 1/2, which gives 1/4.
For a basketballfree all cups run, simply multiply 1/4 by itself twice, or cube it. This gives odds lower than you said at 1.6%, or a 98.4% chance of it showing up at least once.
So it's actually less likely than you thought, so more pain for runners x3
Rest of the post is great though!
Decided to come back and update the times, since looking through Flepty's run for a video I'm working on documenting the Sports Mix All Cups World Record timeline made me realize I had guesstimated the original theoretical times (which I didn't know slipped my mind until the end of this year).
Eh, live and learn. The theoretical times should be updated now.