I'm not sure if anyone has ever asked this one, but what was the Midway doing so far away from Confed controlled space. That just seems like a terrible idea to have a shake down cruise outside of a safe zone. Especially when a ship like the Midway costs as much as it did.
The Midway is in Confederation space when the game starts and then receives a suspected Kilrathi distress call from the H'rekkah system. The decision to jump there was not made by Captain Wilford -- the order came from headquarters and probably marked the effective end of her shakedown cruise.
Per the ICIS booklet, the Midway had started her cruise [at least] 'several months ago' and was well into the 'final phase' (Blair had been observing said phase for 50 days.) They were also almost two months past drawing their first blood, having dealt with a group of Kilrathi pirates earlier on.
In fact, when the first briefing opens you overhear Blair complaining about this exact issue - "Take a look around, most of these pilots are still kids. They're too green... We don't even have a full complement of ships." (Blair, of course, has a strong suspicion about what is going to happen, having already been privy to Paladin's classified briefing on the alien threat.)
Perhaps the Midway was already past the shakedown cruise and ready for action? Or does Confed apply the term differently?
We know that she wasn't past her shakedown cruise at the very start of the game; at their first meeting Stiletto tells Casey and Masetro that "this is a make or break shakedown cruise for us." My guess would be that that status specifically changes at the first briefing, where Commander Drake announces that "as of now, the Midway is on full alert status."
We know of two major builders - Sirius Prime Engineering which is in the Sirius System and Trojan IV shipyards which orbit Mars. We also see drydocks on the Moon in Fleet Action, where the Kilrathi destroy a half dozen Confederation carriers docked there, including the TCS Wolfhound.
I don't think this information is accurate. We don't know the ultimate fate of the TCS Wolfhound, I can't think of any canonical references to any "Sirius Prime Engineering," we never hear where the Trojan IV shipyards are and only three carriers were destroyed in the drydocks at the moon.
The Midway was constructed at the TSY Arcology.