If you use the software without modifying it directly (such as building on top of it, or building something that uses it), then that's allowed.

(IANAL)

Not in the case of AGPL (use over the network and IPC counts as distribution -- presumably proxying the request is insufficient to disable this clause) and even in the case of GPL that's a very problematic position to put yourself on. You're basically talking about invoking a foreign process from your primary process to avoid licensing constraints and that comes with a lot of limitations as to what you can do.

You can modify the GPL program to support more things via IPC but then if that program needs to touch a customer's computer, you have to contribute at the very least those notifications and any related improvements you made to make that possible or any new feature which makes more sense to be in the tool you're calling than your tool building on top.

And last, if you don't modify the software but charge people using it, that's completely allowed.

Yes, but who's paying for that? If it's a server hosting company, they'll pay the hardware rental fee, fair enough. However, you can't reasonably sell that software itself, people will just build it themselves.