Well content/presentation wise you're fine. (except that CoD dig, a bit hokey)
Gameplay wise, there were indeed thing you could have done better even in that first fight. As you've already said you picked the one ship most susceptible to randomness at the start in the whole game, and got a really bad initial start. This would have not been even an issue with most other ships. Being that you have no shields dodge is your only defense at the start besides cloak. So moving a guy from the pilot system was a mistake as was moving the guy from engines. Keep the pilot in his place and the engine room guy (at least for the laser dodge chance if not the beam). Weapon speed boost is insignificant early on with untrained crew member, so move him to the door system, this would reinforce your doors, allowing you to vent the rooms with the mantis more effectively. You would have kept your dodge, and prevented major boarding damage. Then cloaking, it should not be done right at the start but right as the enemy shots, wasting their charge time on misses plus giving you more from the cloak. Loosing a crew member to the mantis was again all on you, unless you are doing a no pause run on purpose this should never ever happen. Trying to put out 3 fires in a 2x2 room and basic doors, instead of venting it, another big mistake. Finally when trying to get away, one dude should have gone to the engines.
Survival was not 100% but it was possible. Yes randomness hurts, but in what you showed here, you dug your own grave too.
The madman encounter is not worth the risk unless you are swimming in resources/upgrades etc or have a way to grantee success with a blue option.
Not exploring sectors as much as possible before jumping: BIG NO-NO.!!