ESnet is not affected by the shutdown. Also, ESnet does not transit all DoE traffic. ESnet is a purpose built science network and if you look in the AS-Set, this network is not transited by AS293 (ESnet).
The plan includes pay for external contractors and security-essential functions, so yes I would assume so (though I don't know firsthand about them specifically). To the best of my knowledge, DoE will continue to run entirely as normal until carryover funding from previous allocations runs out. Couple months at least. After that it basically goes into hibernation, but I expect that communications and network security would be considered as essential and hence would remain funded even then.