If you want me to change House,
Why? Your House and Region are fine, it's simply that time-specific triggers during character creation have to be changed. The specific one in the main rules packet is "Clan War Orphan" becomes "War Orphan". FWIW, there's also specific rules in MW3e regarding tailor text for appropriateness to the campaign. Page 23-24 have an example, so you aren't doing anything rules-wrong by changing this.
And yes, math 98% checks out, though I'm going to ask in the future you format posts so they're a little easier to parse through. The order of operations is hugely important (when something applies to your Attribute thresholds and when EDG is spent, for example), and I'm fairly sure that this helped cause the issue I'm about to address below. Also, in the future, please don't edit posts with dice rolls in them. As much as I hate to point it out, it calls the legitimacy of the rolls into question. Make a new post with the new roll, please.
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The main remaining issue is this:
"Attributes:
STR 5
BOD 5
DEX 5
RFL 4
INT 5
WIL 6
CHA 7
EDG 5 (cost 7)
SOC 4
Total: 48 (2 in pool)
6 total in pool
Changing it since no need for SOC being 6 due to spending edge on lifepath."
This statement isn't correct, and comes down to a mistake in the life pathing order of operations. The attribute minimums are still required; you still attended Sanglamore, so you still have to end character creation with a SOC of 5+. Edge doesn't allow you to ignore minimums set by the path - it allows you to ignore the path restrictions and prerequisites to get onto the path in the first place (MW3e, pg 25 "Edge and Path Restrictions"). In Sanglamore's case, the prequisites are as follows:
Affiliation Lyran Alliance only (Federated Suns may attend prior to the secession of the Alliance in 3057). Cannot have Combat Paralysis or any of the following Traits above the lowest possible level: Disabled, Lost Limb, Poor Vision, Poor Hearing. Character must have one or more of the following: Contact-Free Skye, Well-Connected/Free Skye, WellConnected/LAAF, SOC 6+. May NOT have Stigma/Pro-Katherine or Stigma/Pro-Victor. The character must not have any serious/y "questionable" Events in his or her history relating to Skye (per the gamemaster's judgment) unless the character has or purchases an appropriate Contact who can hide the Event. By spending 1 point of EDG, you ignore all of the above text, and then you can enter the path. THEN the SOC minimum is set to 5. Or to put it another way, the SOC minimum of 5 is not required to ATTEND the Academy; it's the level of SOC score you get as a result OF attending the Academy, so since the Academy happened to you, you have to have the SOC score to represent it.
You'll need to find a Trait Point somewhere so you can shove a point back into your SOC stat.
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The other math issues I was seeing were mostly addressed upthread (Promotion, and Rank, specifically). You incorrectly combined your pair of Contact (2)s into a single Contact (4)...and then sold back half of it. So it ends up being math-correct.
Again, these are tweaks and you don't need to do a full restart or anything.
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To sum up:
1) Fix the time-specific triggers and text
2) Fix the Natural Aptitude issue
3) Find a point somewhere to plow into your SOC stat
4) Find some points somewhere to get into a heavier Mech. Whether unintentional or not, this is an excellent example gaming the system because you're planning on having tbe new PC take over an existing lance, you've got spare Mechs in your lance, and you're planning to immediately seat your Lance Leader in a heavier Mech, and so why spend Trait Points on a heavier Mech in the first place? TBH, I can think of no possible way that this was done unintentionally, given the number of Trait Points sold back which could have been used for this, and so I'm telling you to change it so that a precedent isn't set. If it's not intentional, then change it anyway, so that a precedent isn't set.