Summit is an open science machine (meaning any researcher can apply for time, but it's very competitive so getting an allocation is difficult) and is not budgeted to NNSA for stockpile maintenance as far as I know. It should be running mostly non-classified scientific research, though there is a provision for running sensitive research like stuff related to operating nuclear plants. Sierra at LLNL is the one that is primarily targeted towards classified weapons-related research and NNSA operates independently within DoE.
It's not budgeted for it but I heard that some of the FastForward 2 program money ended up going to ORNL because NNSA needed to do some early stage verification that was prohibitively expensive for them to do on their own but trivial for Summit (since they were already getting the hardware).
AFAIK there isn't a single publicly owned supercomputer in the US that wasn't funded in small part by stockpile maintenance budgets, even if they were never used for that purpose.
Very likely true. I know that stockpile maintenance is at least a tangential concern for all of them, even if it's just to have something available in reserve. Budget allocations are complex beasties, especially in DoE which pursues a variety of missions. The vast majority of work on the ORNL computers is not weapons-related, however.
