Explain to me how who goes 1st on Fleet Arena works?

If two ships are the same speed, then who goes first is a coin flip. 

If the fastest ship on each side is not the same speed as the other, then the fastest ship goes first. 

Speed increases with the GP rating of the ship, which increases with Stars, Level, Ability Increases, and Pilot/Crew GP. 

If your ship is 5 stars and you're facing an enemy with a 6-star version of Punishing 1, then you're very probably going to have a lower speed and always go 2nd. (Of course, if you take Dengar to r9 or whatever, you might be able to overcome your star disadvantage and get the higher ship speed.) 

You can look at ship stats the same way you look at your character stats to see the exact speed of each ship if you're curious. 

As for stealth, it has to do with how this ability is coded: 

Whenever an enemy gains Stealth, dispel it, inflict Breach on themselves for 1 turn, which can't be resisted, and Punishing One Stealths for 1 turn.

For whatever reason -- and I'm not sure why -- this doesn't appear to kick in until P1's first turn. Which means that whichever P1 goes first gets to keep stealth but dispel it on the enemy P1 & XB (if XB is present). I'm not saying that this is correct according to the kits, but it does seem to be how it is working. 

Hope this has helped. 

RNG is CG's employee of the decade.  They also give the AI advantage as compared to players in a few circumstances, and have admitted so in the past.  They have certain ships and characters that get higher priority as targets over others, and so on.  If your capital ship is the exact same as the other, the coin flips and you will notice runs of going second often.  (Just confirm that your capital ship leader is R9 and has all 6E mods on?)

As far as the punishing one, the AI's is stealing your stealth most likely because yours is stronger.  It's a weak zombie in ships, though there might be other issues with it too that aren't working right.  CG's priority for bugs they create in PVP situations is fairly low.  If it was beneficial to players in PVE, they'd be sure to fix it fairly promptly. 

Try an alternate counter?  Check out bitdynasty's youtube channel or his webpage.  He has counters for lots of things, including fleets.  You might just want to use other options to get through..

What’s your ally code? These figures make me wonder if you have the same setup as your opponents. Are these results for mirror matches, or just all of your matches, regardless of what fleet you are attacking?

Its not supposed to be 50/50 in the way you think. Its 50/50 over a single coin flip- your results show it is working as intended in that the result of each flip is not influenced by the results of previous flips.

The issue here is that you are making an incorrect assumption as to what 50/50 means / do not understand basic probability theory.