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The 30-day late payment from 8 months ago is definitely dinging his scores, and could potentially prevent him from getting approved, so getting it removed will improve the scores and his chances of approval. "Much of a difference" is what you define it as, without running a simulator it's tough to guess how many points he'd gain.
If you are qualifying for Fannie Mae/Freddie Mac (conventional) financing, then for student loans not in deferment, the payment that is listed on the credit report can be used to qualify or if you have documentation on the actual payment then that can be used instead (Fannie Mae's guideline). VA will also go by that guideline. FHA & USDA do not use IBR payments and will use 1% of the balance to qualify instead.
Regarding the AZEO & paying off a vehicle's impact to scores questions, https://ficoforums.myfico.com/t5/Understanding-FICO-Scoring/bd-p/ficoscoring would be a better section to post those in.
IMO Do not apply for anything! From my experience, mortgage scores are the most sensitive to inquiries.
I went through closing a home in May and middle FICO MORTGAGE score was 752. Closed on home and applied for 3 cards I wanted and that score went to 699 in 1 month. The FICO 8 version went down 7 points in same time. In veiwing my mortgage scores up to the time of approval the mortgage scores jumped about 30 points on average at 6 months without an application of any of the bureaus.
Good luck!