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The negative impact that a 60 day late will have on your scores and general credit profile gradually fades over time but will continue until the derog falls off of your reports in another 5-1/2 years from now. While everybody's experience is different if you maintain an otherwise clean profile you'll be able to start getting approvals for very good cards well before then, but generally with APRs at the high end of the ranges.
If you haven't done so you can consider asking the creditor in writing for a goodwill gesture to remove the derog; while odds of success are poor it occasionally does happen depending on the creditor. Early exclusion isn't going to start to be a potential option for you for another 58 months.
The new applications will hurt you for 12 months. A slew of new credit denials suggests the applicant may be in dire need of money. Stop applying and let the inquiries clear.
You asked the wrong question about the 60 day late - it will affect your score for 7 years. However, after 24 months the impact on score drops quite a bit and many creditors may no longer consider it a red flag worthy of instant denial.