>I can no longer use Grafana unless I also publish my SaaS under AGPLv3, right?
If you link to it directly from your code then yes, unless you're willing to pay Grafana for a commercial license (assuming they offer one or you can negotiate one with them). If you have Grafana running on your servers but you aren't using the code in your own applications then you're generally fine (the GPL is generally recognized as ending where a process ends). And that's the point of the GPL/AGPL, you get the code for free, with the only expectation being that you pass it on if you make any changes.
If your server runs Linux you don't have to distribute the source code of Linux to the users of the services running on it. If Linux was AGPL you would have to. And yet, you would not have to distribute the source of your closed source software running on the server.
You wouldn't, as system interfaces are specifically excluded as an AGPL enforcement domain. But if the Linux kernel was AGPL, you would have to supply the source code to your NFS clients, since those clients would be served directly by the kernel.
The only difference between GPL3 and AGPL3 is what constitutes distribution. With GPL3 you have to provide source to anyone you provide binaries. With AGPL3 you also have to provide source to anyone who uses the software remotely (eg a web app). Everything else, including the scope of what must be included in the source (modifications, linked software, but not software that talks over IPC, etc), is identical between GPL3 and AGPL3.
More specifically, the only difference between GPL and AGPL is section 13 (and the preamble summary). The rest of the license is word-for-word identical (save for the name).
https://www.gnu.org/licenses/agpl-3.0.en.html#section13
To expand on that, you have to offer to provide the source to "all users interacting with it remotely through a computer network". If you're using Grafana for monitoring your SaaSS app, then your users aren't interacting with Grafana, so you're fine.
If your SaaSS app surfaces a few small things from Grafana to the user, do you only have to open source the glue code, or the entire SaaSS application?
The entire application, to the extent that it's a single work under copyright law. If your app is a "mere aggregation" of e.g. grafana administration and other-service-administration components then maybe not.
If you link to it directly from your code then yes, unless you're willing to pay Grafana for a commercial license (assuming they offer one or you can negotiate one with them). If you have Grafana running on your servers but you aren't using the code in your own applications then you're generally fine (the GPL is generally recognized as ending where a process ends). And that's the point of the GPL/AGPL, you get the code for free, with the only expectation being that you pass it on if you make any changes.