Should I inform my letter writer about a summer school rejection? Should I ask the same professor to write me a letter for grad school?
Yes, you should tell them and yes, it makes sense to ask them again. Likely the rejection wasn't due to the letter they wrote, but to the level of competition for positions in the program.
But, in general, your letter writers should be people who know you and your work well and can make a confident prediction of your future success. I assume that this is one of those people. And informing them might get you an even stronger letter.
Note, of course, that the standards are different for a summer program and a graduate program. And the competition won't be less. Apply to universities that fall over a range of rankings so that you have a better chance of acceptance. If you apply only to top ranked small schools you greatly lessen your chances.
While a summer program has a fixed number of &quot;seats&quot;, applying to several schools opens the number of possibilities.