Suspend Quartus processing

There is no way to suspend it, sadly. Note that you can go to Tools -> Options -> Processing and tell Quartus to run as a lower priority thread. For a single CPU machine this is great, since it keeps things like email/internet very fast, but allows Quartus to run in usually the same amount of time. The only downside is if you run something else compute intensive, like a modelsim simulation, it will completely hog the CPU.

For the second one, Assignments -> Settings -> Compilation Process -> Smart Compilation will try to skip modules that can be skipped. It's not perfect though. You can also go to Assignments -> Design Partition Window and set the top-level partition to Post-Synthesis, which tells it to skip synthesis when it can. I don't think this is available in web edition versions though.

Finally, note that synthesis is now timing driven, so technically a timing assignment should require all modules to be re-run. It's probably fine to skip synthesis in most cases, but something to be aware of.