If I'm 25 and work for the next 10 years to cover the 40 credits required to qualify for Social Security, will my payments be indexed to inflation when I retire?
Yes, sort of, unless Congress changes the laws. Currently, past earnings are adjusted by an "index factor" (to use the SSA term) which isn't precisely the inflation rate, but is similar. It is calculated from average wages for the past years compared to the current year. Here's a link to an example: https://www.ssa.gov/oact/progdata/retirebenefit1.html More detailed information can probably be found on the SSA site.
Under the current rules, yes, benefits are indexed to inflation. However, note that if you have a government pension from work not covered by Social Security, you may face an offset that could reduce the benefits.