Yes and yes and yes.
The ssid of the mac mini will be broadcast. Most offices have a couple of their access points listening for other ssids and trying to figure out if based on signal strength it's from within the office. They probably automatically email someone about it. Whether that person cares is another matter.
The mac being proxied has all its network traffic monitored and/or filtered. If it's NATing for your phone, that will be visible. Again, unclear on if they care. But even if they don't that proxy might be blocking several things that the only phones access, like maybe the ios app store.
IT companies lock down networks because they have resources within them that they can't effectively lock down any other way. That is sad, but if they're limiting what devices can join the wifi that's probably why.
If you can configure the sharing so that each device ends up on a vpn that can only communicate externally (like to the internet) and back, then probably you'll be keeping things secure and can make a case for doing so. EG two phones connected to your wifi point shouldn't be able to talk to each other, nor to the mac, nor to any of the internal ips that the mac probably can talk to.