Use Illustrator without having to use/download Creative Cloud, too.

Creative cloud is completely uneccessary bloatware that limits use. I get that you wish to track licences or whatever else corporate enforces - but the fundamental software experience is highly damaged by the cloud integration. Listen to your users maybe - you'll loose more and more people to Inkscape and GIMP

No.  CC Desktop App is essential for managing your plan services and apps.  It's all or nothing.

It goes without saying, you need a fast Internet connection and a modern computer that meets all the latest system requirements to use Creative Cloud products.  Failing that, your experience will be less than stellar.

Please ensure your computer exceeds the minimum system requirements to run Creative Cloud Desktop App + OS + all other apps you plan on using. Check each one.
- https://helpx.adobe.com/creative-cloud/system-requirements.html

Pay particular attention to Photoshop's new GPU requirements. See links below.
- https://helpx.adobe.com/photoshop/system-requirements.html
- https://helpx.adobe.com/photoshop/kb/photoshop-cc-gpu-card-faq.html

Hope that helps.

Can anyone give some direction on how to install ONLY Ilustrator and working offline with it.

By @princess4ubabe

Either you are online or offline. You can't be online and expecting Illustrator not to phone home. It has not designed like that. However, you can work up to 90 days without internet connection.

And yes, people are having difficulties with Adobe Creative Cloud. But I read similar comments from prior products, including products from other software companies.

But yes certain operations make Creative Cloud fail. I had recently the case that a partner with whom I share data was over his cloud quota and my computer stalled...until I did cut the ties. I also love more the cloud implementation of OneDrive. But I understand the technical rationale behind Adobe's decision not to do offline files. A huge Photoshop file is not to be compared to a small Word file.

You can tune some parameters: disable auto-update for example. Disable synching with the cloud is also effective, not working on the cloud is for sure better. Also, huge libraries can slow down the system. There are many ways to failure.