He sounds, honestly, like a very bright, sensitive child. It's very common for preschool-age children to respond to things that are hard for them -- sensory overload, transitions, a parent's illness -- by revving up their motors and getting themselves all torqued up.
First and foremost, I recommend the book The Out-of-Synch Child.
Also know that, as kids mature, they either outgrow or learn to manage a lot of these issues. For the short term, though, he can probably benefit from all the calming skills you can pass along. When you see him working himself into a tizzy, try redirecting him into something that's either calming or stress-relieving. For some children, the solution is "Let's have some quiet time and read a story." For others, it's "Let's get out your energy and run really fast." You just have to try both and see what works. But the idea is not to wait until a behavior escalates to the point of warranting a punishment. Catch him in the process of torquing up, and redirect him to more productive behavior. If you're able to find a pattern that works with this, he'll start to redirect himself.
In terms of your illness (and I'm sorry, by the way), my observation is that children pick up on a lot, but they often misunderstand. So he may know that you're sick, but his worries may exceed the actual situation. My recommendation is not to deny that anything is going on, but to give him the tools to understand: As in, "Do you remember the nice lady we saw who got to ride in a wheelchair? She could go everywhere everyone else could go, even though her legs didn't work; she just got to ride on wheels. Well, mommy can walk, just like you, but mommy gets tired a little more than some people. So my doctor just says I have to take a little extra rest. It doesn't hurt, and I'm going to be completely fine. I just need extra resting time." Something like that.....
And, finally, you might consider meeting with a neurodevelopmental specialist and trying to get a referral for OT. As the mom of a sensory-sensitive kid myself -- one who's made amazing strides (though he still hates crafts) -- I can say from experience that OTs can work wonders.