ZFR: If not, what are better alternatives then? Use whatever I have on my Windows 7 CD, or should I Install+Update and then only go offline?
Why wouldn't you do that? That is exactly what I would do. MS will keep the Windows 7 online updates available for years to come I think, so it is not like the ability to get the "latest" updates for Win7 will vanish when the extended support ends.
My current main gaming PC (an aging ASUS gaming laptop) is just like that. I have Windows 7 64bit on it (I have reinstalled it clean a couple of times, the last time was like a year ago), and installed Linux Mint XFCE beside it. When the Win7 support ends soon, by default I will keep the Win7 offline (by disabling its networking/wifi), and when I need to go online with that PC, I use Linux. I might still go occasionally online with Win7 if there is some specific need.
I would have an option for a free Win10 update for that PC, but I intend to keep Win7 on it for any games that have issues on Win10 and work better on Win7. I have a separate PC (laptop) with Win10 already, and my future main gaming PC will most likely have Win10 preloaded anyway.
ZFR: Speaking of, would being completely offline necessary security-wise? If I stay connected, but only use it for driver-updates and keep my AV up to date, without any browsing or similar, wouldn't that be OK?
There is no simple answer to that. It totally depends on your other machine/network configuration, what will you do online etc. Like, are you behind a NAT router, are the Windows programs you use (like a web browser, antivirus etc. still actively updated on Windows 7, etc. Microsoft will probably stop giving Win7 updates for their antivirus and IE browser so if you are now using them, switch to using some third-party products like Avira Antivirus, Chrome/Firefox browser etc., as they tend to get longer support for unsupported OSes.
The more critical things you do (like online banking, any services and sites where you log in using username and password etc.), it might be a better idea to do them on the supported Linux side instead. Then again you said you wouldn't use browsers on it.
In general, the longer an OS hasn't received security updates, the more insecure it is. I also have two old retro-gaming PCs with Windows 98SE and XP, and I'd never try go online with them, no matter if someone claims they'd still be secure as long as I update their web browsers etc. (not sure if any web browsers or antivirus still receive updates on WinXP or 98SE). One of those PCs has also Linux, so if I need to download something online directly to that PC, I switch to Linux boot). In the coming years, my Windows 7 will be similarly 100% offline, increasingly so.