Electronic – Finding maximum source impedance for A/D converter (SAM3X8E ARM Cortex-M3)

I did not fully understand the dont_sleep thing, but one thing you could try is do the "main work" in the PendSV handler, set to the lowest priority. Then just schedule a PendSV from other handlers each time you need something done. See here how to do it (it's for M1 but M3 is not too different).

Another thing you could use (maybe together with the previous approach) is the Sleep-on-exit feature. If you enable it, the processor will go to sleep after exiting the last ISR handler, without you having to call WFI. See some examples here.

Regarding the ARM Cortex-M3:

Linux requires an MMU (Memory Management Unit). The ARM Cortex-M3 does not have one. It is impossible to run the mainline Linux kernel on the ARM Cortex-M3.

However, there is a variant of the Linux kernel for MMUless processors called uCLinux.

Linux on M3 Guide

ST's Application Note on uCLinux

However, as others have noted, Linux is unlikely to be practical on the M3. It won't run without external RAM. I believe that even the largest Cortex-M3 parts only have 1MB flash, so you'll likely need extra storage too.